J.H.
Times are tough for a lot of people. Many people are living together multi-generational. This living arrangement can be positive. However, I believe it would benefit you for you and your husband to have your own bedroom. It may hurt your mom's feelings, but I think it would be better for your marriage to have some privacy. Remember living in anyone's home is not going to be easy. If you don't already have a relationship with your mother-in-law, start building one.
Next, I think if your mom could still help with childcare, she will not feel left out of the helping you situation and still stay connected to your daughter. Don't feel too bad about how your daughter is attaching to your mom, because it happens in all childcare situations. That's why in an ideal world one parent could always be at home, but we don't live in that world, so we do the best we can. Don't let your mom make you feel bad. And don't listen to her about your husband. If she asks why you are leaving, tell her you need the privacy and you need more emotional support not husband bashing. However, I also think that if your husband is working part time, he should be taking care of your daughter when he is not working. That may be difficult at any place you are sharing a home with another woman who may want to just take over for him, but you'll need to talk to him about how important it is for him to be helping to raise your daughter. Remember here that he will do things differently than you would and that's okay and good.
Another thought...has he looked for a job that has a shift other than your's say swing shift. He may have to look for work that is other than what he is used to. (I know the Sutter Foundation in Northern California always has jobs posted for swing and graveyard shifts of janitorial and food service and when I was looking on line the starting pay wasn't bad.)
I'm sorry that so new in your marriage you are having to go through this, but be strong and your marriage will strengthen. You'll be able to look back and feel good and proud about how you made it through these tough times together.
Hugs,
J.