Hi Maddy -
<<<HUGS>>>>
Along with everyone else's great replies, one of the most important things I can suggest for you to do is to get with your husband to decide and implement some firm boundaries. So often in situations where the children are shared, the boundaries between the adult relationships are so skewed that it becomes unhealthy for all. Many a second marriage ends in divorce over uninformed or much-needed boundaries that never found their way to implementation!
For one, you and your DH need to decide on acceptable methods and times of communication. This usually need not be more often than at drop-offs and pick-ups. If there needs to be further communication, decide on a time that works best for your family/marriage and communicate that to the BM. If she's like my BM, she will likely challenge this, in action, if not in word. That is fine. That is where yourself and your DH come together to impose these boundaries. Get caller Id if you don't already have it and do not pick it up if calls come in outside the communication parameters you have outlined. If you are worried about genuine emergencies, the 10 extra seconds it will take to check an answering machine message will not make or break the outcome of an emergency and you will soon find out that 99.9% of the interruptions are not only non-emergencies but completely unnecessary and could have waited until a drop-off/pick-up, at very least.
Cell phone usage is a common area where boundaries need to be created. There is nothing more frustrating than having a night out with your husband only to be interrupted by his ex-wife! I suggest changing any cell phone numbers she might know and not sharing any future numbers. As long as she has a way to access your DH for important parental information (i.e. your home phone number), there is no reason why she needs to have instant and unlimited access to either of you.
Additionally, I *highly* suggest you cut off all email/IM contact completely. It is just not necessary to have such constant communications going between you all when things are so high-conflict. This will go SO far to alleviate your stress.
Finally, you should never be part of these communications other than acting as a consultant to your husband! You have no relationship with your husband's ex-wife - only he does and you should encourage him to do all the talking and advocating for his kids with his ex-wife.
When it comes to the kids communicating with their Mother, it is important to leave those lines of communication untouched, in my opinion. If the kids ask to talk to their Mother, they should be allowed. However, if the Mother uses your new boundaries as an excuse to overly-impose on the children via phone/email/IM, it is perfectly acceptable to set reasonable limits on phone calls as well. After all, it is your parenting time, not hers. Likewise it is important to respect and not to impose on her parenting time.
As far as the denial of visitation, it sounds like you are all doing the right things. Just make sure you are documenting EVERYTHING and keep those records indefinitely! Keeping records in pen is best but a computer will do if necessary. Every refusal of visitation should be recorded with details about the circumstance surrounding the refusal. Any threats should be logged as well. In some states, it is legal to record phone calls and that is a great way to document her lies. Be sure to research the laws in your state.
Good luck and hang in there! The lesser contact you have, the more you will be able to handle the stress when it inevitably occurs!