hard to answer without seeing sample questions that seem befuddling...

but at the least here are a few brief comments:

in mathematics, the easiest problems are those about the small and finite (including parts of discrete math). why? because you can compute every case by hand!

somewhat harder is problems about the infinite: but still, an infinitude (or something like it) yields to techniques from calculus, analysis, etc.

the hardest problems are those around the big and finite (again, including parts of discrete math). why? because you neither have the analytical techniques [immediately] within reach nor can you feasibly check all cases by hand.

i mention this to open because, frankly, discrete mathematics is hard. you learn early on that 52! is the number of ways to rearrange the cards in a standard deck, and, although this may not seem like a tough combinatorics problem, actually grappling with such a large number can be frightening. for example:

"it is quite likely that any given configuration achieved through random shuffling has never appeared before in the history of shuffling!"

well... back to your initial question:

Any tips on succeeding in discrete math?

do a lot of problems. "there is no royal road to [discrete math]" and this is really the best way to strengthen your abilities in combinatorics, graph theory, etc. one specific idea, though, is to take some of the problems that you have solved and see if you can put similar ones together. right now, you write:

I feel like I have zero intuition for when to use each method...

you could try to develop some intuition in this direction by gaining practice with sorting-by-method for problems that you already know how to solve. can you, for each of the problems already solved, write down a name for the method used? (e.g., mathematical induction/well-ordering principle, pigeonhole principle, bijections, combinations/permutations, weighting edges/vertices, computing small cases and looking for patterns [Fibonacci, Catalan, etc], working backwards, drawing a picture, solving algorithmically)

also, once you've found supplementary problems, you could try giving them to the tutor and asking her how she thinks about them. to hear someone articulate their own thoughts in how to broach a problem they haven't already seen might give you some insight into how they work through such things. one note of caution in this direction: ask beforehand if she'd be cool with this, and indicate that your goal is to gain the aforementioned insight even if she doesn't solve the problems. if i was tutoring someone and they just pulled out a bunch of non-routine problems that they'd found online and said solve these in front of me so i can see how you think i'd wonder what their motivation was (and it could come off as quite confrontational). so, run such a suggestion by your tutor (or anyone else -- professors included! -- who you think is an able problem solver in this particular realm) carefully beforehand.