If I land on an asteroid and my movement template partially crosses the same asteroid again on my next move, do I have to roll for damage again, or is it assumed that I already paid for it? Thanks

Again. And each turn you repeat it.

So don't do it.

Each turn you either hit an asteroid or you don't. While you may get to ignore any asteroid you are sitting on when you start the turn you do not ignore any other part of that rock so if your template overlaps it again, or you land back on it again, you suffer the effects again.