So Pearsons Chi Squared Test tests whether differences in observations between categories (say A and B) occurs by chance. The way I interpret one of the assumptions of the test is that A and B must be drawn from the same sample or samples of the same size.

I need to do a similar test, but the samples that A and B are drawn from are different sizes (they are both drawn randomly, so there shouldn't be sampling bias). A and B can't be observed at the same time because of the technique. My initial thought is that I can use Fisher's Exact Test, but I don't know how this would be set up (not that familiar with the exact test families). Or is there a completely different test or adjustment for the Pearson's X2 test?