The simple rule in English is that you can always use the indicative: if Hamlet was still alive...

Stylistically, some people also have the option of using the subjunctive (were) in the way previous posters have described. But this is not "correct English" as opposed to not using it. It is a register that some speakers were brought up to have and others weren't. To repeat: 'was' is _always_ correct.

I make the distinction in more conservative style, and tend not to in less formal style.

'May' vs 'might' is a bit different, because 'may' is becoming less used generally, so 'may' in the sense of 'might' sounds like a conservatism, and in that case, if you're going to be choosy in words, you should choose the more logical 'might'. When I catch myself writing 'may' I almost always amend it to 'might'.