Does anyone have any tips on how to not cramp up on sit-ups during the APFT?

Train better by checking your form. It sounds as if you are crunching as opposed to rolling out and pulling up. Train to do a series 25, 30 correct without pain and add 3 to 5 at a time until you are able to do as many as you like. Whatever you do, do not slam down and jerk back up, you will likely raise your tail bone or button in the effort effectively failing the test. Train to standard and build on your entire abdomen, including your lower back and obliques. Work out with a peer and check each other’s form. Thank you for your service.

I never maxed sit-ups until I stopped doing them. I found hanging from a bar and doing knees to elbows was better, as well as GHD situps if you have a GHD to train on. Stronger abdominals mean less reliance on hip flexor and less cramping.
As for form - tight on the rise, relax on the fall. Breathing pace is key.

Could be anything...I guess try hydrating more especially the week or two prior to the test, stretch more, and make sure to do a wide variety of ab exercises, not just crunches.