I know this is a hard decision for you, and I can tell you're really upset and stressed out about this. It's unfortunate that your ex is using the children as pawns to upset you even more. I don't understand people who divorce and use children like this. It's childish and it's cruel. And it certainly is not doing the children any good.
I hope you have actually MET this woman that you're in love with, since you say this is a "long distance" relationship. Some people get in these relationships and don't even meet until they decide to move in with each other.

I hope you know what it would be like to live together, if that's what you intend to do. I hope you know her "quirks" and habits, good and bad. These things are important to know before you move in. Who knows -- she might drive you insane!!!
But if you think that she's "the one" -- I say go for it. Yes, I know you won't see your children as often, but you can't stop living. Just make SURE that you make EVERY EFFORT to see the kids as often as you can, and do NOT skip any visits when the times come. I don't care what the excuse would be, don't skip visits!! These kids are going to need you even more since your visits are going to be less often.

And make sure you remember all birthdays & holidays with cards, calls, etc.
I wish you the very best. I hope things work out between you. God bless and keep us posted, okay? Hugs, Lee