I am wondering if a crossmember from an 88 Formula will fit a 82 Trans Am. The 82 has a 200C, so I was thinking about switching trannys. I am wondering if any holes will need to be drilled, or any other modifications.

No, the crossmember from the 88 formula will not fit your 82 trans am. The hole locations on the frame are different, it would probably be easier to modify your current crossmember. I am doing the same swap this week, I ordered the crossmember from Spohn its about $100.

Yep it doesent fit, I made the mistake of cleaning & painting the one I got from the junk yard $20 bucks down the tube.
I used the old 200c crossmember. I slid the crossmember back and with a little enlarging of the front two holes was able to to mount it in the back nuts, two quick spot welds and it was done.
Its held up for a year so no complaints. The 700R4 is well worth the effort.

Edit. BTW the 200c tq arm doesent fit right either, your going to either have to get a 700r4 tq arm or chop about 2 inches off the 200c one so you can clear the sppedometer housing.