That's a rough thing to hear from your H. Validate but don't confirm. "That must be really difficult for you. I'm sorry you feel that way." And, whatever you do, don't defend yourself or try to correct him. That will just cause him to fix that notion deeper. He will have to fight this battle himself. You are doing everything you can.
You are right to wonder if you can go on with someone who doesn't believe you. Again, don't try to be believed, as much as you want to and as painfully frustrating it is to be unfairly judged - especially by people who have a number of problems and things to answer for regarding the damage they are causing your family. Just detach from it all. The ILs, your H, the comments. It is not your problem. You move ahead w/ your & your kids' life while H figures out if he wants or is able to deal with his internal and familiar problems. That at least protects you from getting drawn into their little world of dysfunctional anger.
Your boundaries w/ your ILs are important. The key one is to not let your MIL mistreat you or disrespect you in your home, in front of the kids, or even if it is just the two of you. You can't do much about what she says when you aren't there, so don't worry about it. Another boundary is to ask H not to pass on what MIL says about her, and especially if the kids are around. Tell him that she feels the way she feels, but this is her problem and she needs to deal with it herself.
Whatever you do, unless there some obvious issue of harm to the kids, do not get between the kids & their grandparents. You are the only adult in the family system you operate in. Take pride in that, and do what is best for everyone, even if some of them don't really deserve good treatment from you. In the long run, the only thing you control is your words & actions, and treating those who don't reciprocate well is for your sake, not theirs.
You've got some stamina, and you are doing great DBing. You just may not have enough on the other side to work with. I'm glad you are facing that reality, as that takes a lot of courage.
And, you need men & women in your life that you can have a deep meaningful conversation with IRL. If you have women friends already, great. But having both more men & women in your social circle with whom you can have these, sounds like something important to you and worth thinking about how you will move toward that goal, regardless of whether or not your M turns around. You know from this board that we're out there. You don't have to date or have affairs to have these kinds of friendships. There of course will be dangers. I suspect you would really feel drawn to someone like this right now - I'd take a wild guess that even though you don't know what any of us look like - you've found at least some of us attractive, or at least wondered... It is normal, btw, and healthy. It's a sign that you care for yourself and want something good for yourself. It is just that right now, that might draw you in like the oasis does to the thirsty traveler.
Hang in there. You H & ILs sound like they are an incredible test of human patience. I'm amazed you have been able to respond to them as well as you can.
Me: 50 W:43 S6, S3 M: 12 yrs. T: 17 M is bad & Not happy Bomb Mar '14 S 5 Feb '15 D Bomb 13 Apr '15 (but "no hurry") DB Coach May '15 Wants proceed on D Aug '15 Starting 1-on-1 negotiations Sept '15