If you think you might have gotten something wrong, but you aren't sure, don't change it. Go with what your first thought was, and you're more likely to get it right. Mildly uneasy (yet unspecific) feelings that you may have got it wrong are more likely nervousness than a problem with your answer.
Sometimes you'll do something wrong, and then catch it and know for sure it's wrong. Go ahead and fix it, you're probably right about being wrong.
I never check any answers until I'm completely done with the test, but then I check more or less everything. Some things are tedious calculations or would take as long to check as they would to solve initially, I tend not to check these, unless there's tons of time and I'm not quite sure I got it right. Sometimes finding the right answer is much harder than checking that it's right, and I'll definitely check all of these if I have time.
I don't turn it in unless I've got a warm fuzzy feeling about every problem, or I know that I probably messed that one up, but that I definitely can't fix it with the knowledge in my head. (Or if time runs out.)
Sometimes there will be a hard problem that I don't think I can do (or finish), but I'll keep trying unless I'm absolutely sure I just plain don't know it. Mostly, I'd rather spend more time on trying to get a hard problem done than checking the rest of it, since the rest is probably fine, and chances are probably better that I'll get somewhere on the really hard one than that I made a mistake on an easy one that I'll end up catching and fixing correctly.
Never leave a question blank unless you run out of time or you absolutely have no clue whatsoever. If you show that you know some stuff, but not enough to solve it, you'll still probably get partial credit.
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"A dead thing can go with the stream, but only a living thing can go against it." --G. K. Chesterton