Your MIL and SIL sound horrible.
Your baby is not a toy, you don't have to share him. You especially don't have to expose him or yourself to people who see fit to verbally bash you when they want attention. If you they can't be pleasant and supportive of you as a new Mum (case in point- their actions to date!) then they shouldn't be around you or the baby at all.
DH needs to step up and manage them. He needs to tell them their behaviour was unacceptable and that he doesn't want their tension around his new family at this time.
There really needs to be consequences and a clear message that this behaviour is not on, or she will keep doing it to get her way. She doesn't get to pretend everything is dandy and be welcomed into your home. She dropped a series of emotional bombs on your family at a time when you were at your most vulnerable. It's incredibly selfish and destructive behaviour.
I suggest you move this question to the Relationships board, which has great advice about managing family members like this.
I also recommend ordering a copy of Susan Forward's book 'toxic in laws' and reading the Captain Awkward blog.
My PIL threw tantrums in the weeks following my first child's birth as well. It was my first real clue as to how controlling and cruel they were going to be. I was too tired and overwhelmed to deal with them properly. But looking back I see it was a crucial time to communicate boundaries and acceptable behaviour, but I missed it. I was focused on keeping the peace and trying to calm them down and make everyone happy. The result was they felt empowered and entitled and their behaviour just got worse and worse.
It's sadly common for new grandparents to do some version of this. They feel a loss of control and relevance when their son/daughter have a baby- because they aren't in charge anymore, they are spectators. They aren't the focus of their DC's attention. Normal healthy people are happy for the new family, proud that their child is independent and doing well, pleased to take a supportive role at this stage of life.
But that's not your In laws. They are trying to hold on to their power through the use of emotional blackmail.
Show them it will not work.
If they are kind, calm, supportive = time with GC
If they are cruel, dramatic, demanding = 'sorry Mum, that's not on. Let's try another meeting in a few weeks when you've calmed down.'
Keep her accountable.