There may be another reason behind OW's speech to your son's about the "lying". She obviously has some kind of agenda that is supported by your XH. You mentioned that you have a court case coming up regarding custody. It sounds like she is preparing your sons for her arguments in court. If she can get them to tell a judge that you've lied or you're mean, then she may think that she can "win" whatever it is she wants. I would immediately contact your attorney and she cannot be allowed to alienate your children from you.
As for the texts and calls, I would inform your XH that he is welcome to call but she will not be speaking to the boys. As soon as she gets on the phone, end the call. You've already stated that you bought the phone so that your boys have access to you because your XH was not answering calls. Let them know that when the boys are at your house the phone will be off. Your attorney should have no problem with that but you can run it by him first if your more comfortable.
I would also save everything she sends and forward to your L. It would also be to your benefit to document all of your XH's contact for right now since he's trying to say you're denying him. Keep a log of nightly calls and visits.
In the states, you don't have to exchange children at the house. There are agencies that will monitor the exchange if needed or you can specify a neutral spot. That might help cut out the grandstanding by her.
My first H remarried a real gem. She and I have gone round a few times. But when I found my voice, she backed off a little. It will take some time but you'll get stronger and will make all the difference.