The children aren't yours.
Quite a few Dominican men have no problem walking away from their own children, sometimes passing them on the street without anything more than a casual, passing glance.
Apparently the father/s of these children have done the same. If they don't feel bad about not supporting these children, why would you, especially if you're being threatened with legal action?
Think about that. How many children do you know in the DR where the father is never around?
If she is threatening you, which it sounds like it is (maybe you'd like to fill us in on those details, it would help us to know the whole story) maybe you'd like to re-arrange your travel plans and just go somewhere else until this nonsense blows over.
The world is a big place. If you are looking for fun in the sun, there are many other places you can go to enjoy a stress-free vacation experience.
That would be the best way to put this situation in the rear view mirror.
And you need to stop taking her calls. She is playing you for a pendejo, and your attachment to these kids is being used against you.
Foreigners with a heart of gold are seen by people like your ex-girlfriend as a sucker, and in Dominican society there is no quarter given to suckers.
Regardless of what the law states, the second you land you could be in for problems. That's how it is in such a corrupt environment. If your ex-girlfriend has any friends that are cops or military personnel, these people could set up a basic scenario where it would be easy to extort money from you, and the law won't be there to help.
You could also be hit with an "impedimento de salida" and not be able to leave until the situation is cleared up. Why bother with all this drama? If these kids are being raised in such a household, they'll only grow up to be tigres or cueros anyway. So forget the school uniforms and the sneakers and keep it moving in another direction.
So my advice to you is, STEP OFF.