An interesting development is people who don't think these two things are enough! Mostly in continental philosophy, but a bit in analytic as well. I haven't seen the various currents all summarized in one place, so here's a sort of narrative annotated bibliography (for my future reference, if no other reason).
A big pushback against critique as the dominant (sometimes only) approach to philosophy and scholarship in the continental sphere started in the early 2000s, largely from Graham Harman and Bruno Latour. Harman's 2002 book, Guerrilla Metaphysics, advocated, as Robert Jackson nicely summarizes it, a "philosophical style of fascination rather than critique". In Harman's view, critique has its place, but "there is a sense in which the great thinkers are always far more childlike and gullible, far more involved with some mesmerising central idea than all of the wary, uncommitted, replaceable critics".
Latour's 2004 essay "Why has critique run out of steam?" took critique head-on, wondering if it was still doing anything useful, or perhaps even actively being harmful—he pointedly brought up the sort of conspiracy-theory version of critique popular on the American right, which distrusts all appearances and always sees shadowy hidden forces at work:
Of course conspiracy theories are an absurd deformation of our own arguments, but, like weapons smuggled through a fuzzy border to the wrong party, these are our weapons nonetheless. In spite of all the deformations, it is easy to recognize, still burnt in the steel, our trademark: Made in Criticalland.
He goes on to argue that it's not just that critique is being popularized and done badly, but that it isn't addressing the right things anymore. One problem he nicely analyzes is that critique seems to proceed via a dual set of gotchas. In critical gesture move one, the critic unmasks a naive belief in concepts like "nature", "god", "poetry", and what have you, showing that people who claim that those things actually cause anything are merely projecting; it's actually the people themselves who are causing things, and then projecting other causes. But as soon as the hapless subject thinks this means they have any free will, in critical gesture move two, the critic unmasks as naive the belief that agents can do anything of their own accord, because in fact society, capital, and other factors are at work, determining their actions. Either way, the critic wins! But it seems, nobody else has won: we get no new concepts, no things, just endless critique.
Well, if not just critique, then what? How about positively arguing in favor of a position? That's the traditional answer in analytic philosophy, and has been taken up recently by some people in the critical-of-critique portions of continental philosophy. For example, Quentin Meillassoux's After Finitude is notable for proceeding largely by a style of deductive argument, despite otherwise being situated more in continental philosophy.
One problem is that, unless an argument is so unassailable that the point is irrefutably proven, you end up with a lot of arguments for a lot of positions, some conflicting. How do you resolve them? Now you're back to... critique. You find flaws in arguments, construct counterexamples to claims, challenge unstated assumptions, and so on. This can be useful, but some philosophers feel that it still isn't capturing everything that's worth doing. As Harman says somewhere (I can't seem to find the source of the quip), Plato wasn't a great philosopher due to his arguments containing fewer logical errors than the average philosopher. Something else is at work: something about the fertility of the concepts he introduced, the thinking he catalyzed, the novelty of the broad outlines of the arguments (errors notwithstanding), etc. (Harman explicitly disagrees with Meillassoux on the primacy of deductive argument in Prince of Networks, pp. 167-77.)
Harman is influenced in this by A.N. Whitehead's famous comment that philosophies are abandoned, not refuted (Process and Reality, 1929):
It has been remarked that a system of philosophy is never refuted; it is only abandoned. The reason is that logical contradictions, except as temporary slips of the mind—plentiful, though temporary—are the most gratuitous of errors; and usually they are trivial. Thus, after criticism, systems do not exhibit mere illogicalities. They suffer from inadequacy and incoherence.
A related idea came up in the 1990s in analytic philosophy. In an influential 1996 essay, Peter van Inwagen noted that there was rarely consensus within analytic philosophy on almost any question of importance: on any question where multiple coherent positions have been developed, you can find intelligent and thoughtful people holding each position. If philosophy is about argument and refutations, how is this possible? It didn't appear to be the case that the philosphers holding opposing views were either stupid or ignorant, since there are many disagreements between philosophers who are each well aware of the arguments opposing their position, and widely viewed as quite intelligent thinkers:
How can it be that equally intelligent and well-trained philosophers can disagree about the freedom of the will or nominalism or the covering-law model of scientific explanation when each is aware of all of the arguments and distinctions and other relevant considerations that the others are aware of? How—and now I will drop a broad hint about where I am going—how can we philosophers possibly regard ourselves as justified in believing much of anything of philosophical significance in this embarrassing circumstance? How can I believe (as I do) that free will is incompatible with determinism or that unrealized possibilities are not physical objects or that human beings are not four-dimensional things extended in time as well as in space, when David Lewis—a philosopher of truly formidable intelligence and insight and ability—rejects these things I believe and is already aware of and understands perfectly every argument that I could produce in their defense?
Again, something else is clearly going on: David Lewis doesn't reject Van Inwagen's positions because of a lack of arguments in their favor, or due to unfamiliarity with those arguments. Perhaps it might be different if there were truly irrefutable arguments in their favor such that no intelligent person could fail to believe them, but it appears that such definitive arguments are rarely actually constructed.
If neither arguments nor critique suffice, what else is there?
In analytic philosophy, the tentative approach has been to treat it as a new problem, the "epistemology of disagreement". Once the arguments and critiques in favor of and against a position have all been sorted out, and still aren't decisive, we need some new theory of what to do with and how to explain the result. Do we have a free personal choice of what position we want to believe in? Is there a correct way of weighing evidence? Must we remain agnostic as regards all unsettled questions? If epistemic peers disagree, is it possible they possess some sort of uncommunicable insight? And so on.
In continental philosophy, the response proposes to more fundamentally change how philosophy is done, instead of adding a new theory on top of the argument+critique substructure. The general theme is that philosophy should, in one way or another, build things.
In Guerrilla Metaphysics, Harman suggests that, rather than only critiquing, deconstructing, and refuting, the philosopher might take carpentry as a better metaphor: perhaps philosophers should spend their time looking at how things are made, fit together, and relate to and affect each other. This "carpentry of things" has since spawned a small cluster of approaches, ranging from Levi Bryant's "onticology" to Ian Bogost's idea that philosophers should go so far as to make non-textual things (things that aren't books or essays) as part of their philosophizing, an idea he will apparently expand on in his forthcoming Alien Phenomenology.
Latour, after declaring critique to have run out of steam, suggests a replacement adapted from Alan Turing, of all places, playing on a pun on the word "critical". Turing writes (in "Computing machinery and intelligence", 1950):
One could say that a man can "inject" an idea into the machine, and that it will respond to a certain extent and then drop into quiescence, like a piano string struck by a hammer. Another simile would be an atomic pile of less than critical size: an injected idea is to correspond to a neutron entering the pile from without. Each such neutron will cause a certain disturbance which eventually dies away. If, however, the size of the pile is sufficiently increased, tire disturbance caused by such an incoming neutron will very likely go on and on increasing until the whole pile is destroyed. Is there a corresponding phenomenon for minds, and is there one for machines? There does seem to be one for the human mind. The majority of them seem to be "subcritical," i.e., to correspond in this analogy to piles of subcritical size. An idea presented to such a mind will on average give rise to less than one idea in reply. A smallish proportion are supercritical. An idea presented to such a mind that may give rise to a whole "theory" consisting of secondary, tertiary and more remote ideas.
This new kind of "critical" mind takes ideas and multiplies them into other ideas and theories, rather than taking in ideas and dismantling them or unmasking them as naive. Latour expands on this view in his Compositionist Manifesto, which argues that we need to get beyond critique's idea that we're going to "discover[] a true world of realities lying behind a veil of appearances" if only we debunk and unmask vigorously enough.
Follow-up: By way of Ian Bogost and Graham Harman, an observation from Richard Rorty along similar lines.