Find a local Mens/Fathers Support group ***TODAY*** in order to get access to vital resources you'll need (including a good attorney). As you have already found out, since you are a man, the law is NOT on your side. Just be warned that this will be a tough uphill climb and no matter what resources you put into it there is still a chance you won't win. Thank the feminist groups for seeing to that, using half-truths and out-right lies.
Your STBX sounds like the b*tch from hell, which can sometimes work in your advantage, *IF* she goes a bit overboard with her antics. Again, expect a VERY tough battle with a LOT more accusations flying to which you'll find you are guilty until proven innocent - regardless of how you think it should be in the USA today. If you think the past allegations are crazy, you haven't seen anything yet from this woman. Expect for the worst - the ABSOLUTE worst - and pray for the best.
And, since you have kids, the uphill battle will probably get exponentially harder.
Edit: The main thing to do is to NOT go overboard with your emotions as much as it pains you. She WILL use any outburst, argument, discussion as fodder for her case. If you HAVE to talk to her do so with an unbiased witness, if possible. And by all means document *EVERYTHING*.