What a mess.
OP, You and your husband sound like wonderfully kind people.
I'm so sorry that this long awaited pregnancy of yours has been complicated by such stress.
Because this is stressful.
Great that the baby has arrived safely.
However, this could be a very stressful time having a new baby living with you, and then bringing your own home and naturally wanting some privacy.
I also feel for your mother as she quite understandably doesn't wish to go back to the baby stage, is resistant to it and sounds very annoyed by it all.
The result though is that she has handed responsibility onto you and your husband.
This is a huge burden to hand onto a first time mother.
I really think that it would be very wise to explore as many options as is possible.
Your sister's situation is IMO a child having a child.
This is underlined by you trying to figure out what is best for her.
Your own relationship is very important to protect.
Don't sacrifice it at the alter of taking on the responsibilities of your mother and father.
Wishing you the very best.