Her behavior sounds jealous, resentful, and not mature in a way that conveys a sense of confidence. Your reactions also don't sound mature in a way that conveys a sense of confidence. That's causing a lot of friction that probably doesn't need to exist.
Your husband sounds like he's just trying to keep the peace. Their long shared life together won't ever be truly over, as long as they are parents to kids who are still alive. So he needs to maintain a civil, and ideally friendly, relationship with his ex. When you think about it, that's a far better option for the kids and you than the two of them sniping and fighting over everything.
It's not easy to deal with a jealous ex-anything. But one of the finest ways to rise above it all is to… well, rise above it all. Be blissfully unaware of her innuendos and jabs. Don't stoop to judging her for her past failures – we all have them. Smile and be friendly when you have any contact with her, but don't be drawn into feeling bad for anything she wants you to feel bad about, or she clearly "wins" that round, in her own universe. That fuels the next round of sniping and cattiness. In the great arena of human emotions, we're only a victim if we believe we are.
If you are consistently calm, kind, perhaps even emotionally generous with her (not physically, like flowers), she will eventually run out of fuel. And until she does, you'll simply look to all the world like a bigger soul. The real beauty of this approach is that, if you do it consciously and for your own and your family's good, you will feel like a bigger soul!
Blessings to you all. I wish you success.