I'd say both of you could look at life through the eyes of the other, and since you are the one who has asked, I will begin with your side.
You don't let her babysit alone because you want to spend your time with your daughter, so that means you never go out with baby's father alone and enjoy yourself as a 20-year-old should. Are you on your way to living your life through your daughter, as MiL is through her sons?
Let her have her granddaughter to herself for say one evening a week and get out there and have fun. You will be more relaxed about being a parent and she will not be so lonely. Can you imagine what your life would be like if you did not have either the man you love (she sounds out of love with hers) or the children you devoted your life to? And then your son gave you a beautiful granddaughter whose mother resented you loving her?
She took the trouble to change her shift because she thouight it would help you, and give her time with baby, who she wants to have a little bond with too, don't fall out with her over that.
If you give understanding, you will get understanding. Talk to the woman and tell her how much you love your child - add "as you love your sons - as mothers do" but that you see that she loves her too, and that you can help one another - you can get out sometimes and still be young, and she can enjoy having a child to look after again.
And if you can get somewhere to yourselves, do so - but still keep in touch with his mother, and trust your treasure to her on a regular basis - you will be glad of her when never going out anywhere starts to drive you round the bend and you and he are in danger of drifting apart because of it.