I am totally opposed to grandparents rights. Our children are our children. How would OP have felt if her parents had behaved as she has?
I do have every sympathy for her plight and can understand how upset she feels.
My feeling is that nowadays too many grandparents get far too emotionally involved with their AC's lives. Constantly ringing them, thanks to mobile phones or, thanks to cars, constantly arriving on AC's doorsteps, and almost assume that they are virtually 3rd parents to their grandchildren. I
In the past if AC's moved to the otherside of town, grandparents visiting meant a 30 minute walk or two bus journeys, telephoning meant walking to the phone box with a handfall of small change, so apron strings physical and metaphorical got cut quite firmly when people married. Older women developed new lives which included children and grandchildren, but also included, friends, the church, WI etc.
If OP could just stop crying, especially with the child, take a break, apologise to her daughter for her behaviour and ask in an adult way whether they could negotiate for the grandmother see her DGD, perhaps once a week for an hour or so., she is more likely to be successful than behaving as she is.