The difference lies in the qualifiers, "just" and "try." Without them, the choice becomes obvious (at least, to me.)
Edit. ('what I'm trying to suggest here is that the original versions seem best with might, but by eliminating "just" and "try," may now becomes more acceptable.
I'd definitely prefer "might" in the first sentence, but I don't think using "may" instead is wrong.
Sentence 2 is a different story. It is apparently intended as a suggestion, and using "may" instead of "might" to make a suggestion seems completely wrong to me.