It is not easy being a single mother much less with the added jw crap to go along with it. That being said, your job as a parent is to protect your child. Protection means physically, spiritually, mentally and naturally from sexual abuse which is a problem within the jw cult.
It is not easy to clean the house, do errands, etc... with a child as a single parent (or even a married parent) but millions of parents do it every day. You need to stop relying on your parents/family for anything if you want any leg to stand on at all regarding the jw issue.
Find a way to move out of their place even if you do pay rent they are still in 'parent mode' and you are still depending on them.
If they want a relationship with your child it should be under your strict supervision, never, ever leave a child alone with a jw, any jw, including your parents. They WILL take any opportunity to attempt to brainwash your child further confusing a young mind. Let them know in no uncertain terms that if they want a relationship with the grandchild you are a package deal and where one is welcome so is the other. Furthermore there will be no talk of anything jw related at all.
Please protect your child.
I would suggest you find a group of young mothers to befriend and trade babysitting services, advice and friendship. Try 'Meetup' and find a group you may be interested in or a play group to get to know people.
If you have to move across the country and start over to protect your child, so be it. Start thinking of alternative options.