Obviously in the medium - long term you need to be looking at alternative childcare and perhaps a career change. If you work out what you want to do about these areas and start working towards them, hopefully you'll feel less trapped.
In terms of how to deal with her whilst the situation remains the same, I'm not an expert but I'd start with trying to get some counselling for yourself and in the meantime get some books out of the local library on assertiveness.
You don't need to listen to her when she is saying things that make you feel bad. You can walk away. I know that's difficult because I'm a people pleaser too and you feel like its rude, but its not - its protecting yourself which is important.
You could reflect her comments back at her, acknowledging her feelings but without accepting responsiblity for them. i.e:
NSDM: We never see you and DGC.
You: You wish you could see us more even though you look after him twice a week.
NSDM: Yes
You: That's nice.
NSDM: So will you come around this weekend?
You: No. (or No, we have plans. if you really can't stand no as a complete sentence!)
NSDM to DB: We never saw Lulu and DGC at Christmas
DB: Oh that's awful.
You: (laugh) Mum you obviously can't get enough of us, we were round Christmas Eve, Christmas Day and Boxing Day!
Stick to the facts. Her feelings are not your responsibility. The rest of your family may well have a reasonable idea of how manipulative she is and don't believe what she says, I'd try to build up a relationship with them seperate from her. If they challenge you on any of her rubbish then stick to the facts about the situation (like with the visits over Christmas) and say that its a shame mother feels like that (but don't say anything that suggests its your responsibility to change how she feels).
Obviously if they try to say it is your responsibility to make your mum feel better then maybe they aren't good for you either.
My parents both suffered from (at the time undiagnosed) depression when I was growing up. I also felt/feel responsible for making them happy and am a people pleaser. Its difficult to break out from that role. But you really can't make her happy, it just destroys your own peace of mind to take on that responsibility.
Your DH sounds rather sensible realising how awful they are, maybe he can help support you in either reducing contact or in trying to minimise its affect on you.