Didn't you say she'd had several homes in her short life?
Yes this is her third home.
Be aware -- as time passes you may see a different dog. The first two -four weeks are the honeymoon and you may begin to see why she was tossed around. The hyper submission may simply have been what she did after she was scolded and got her positive stuff and she's got all the positive stuff out b/c she wants to stay. She's probably pretty scared of screwing up.
I hope this isn`t the calm before the storm
Pay attention to the action/reaction sequence. Does the dog initiate certain behaviors in humans (such as laughter or belly-rubs) by doing this? Is the dog rewarded for manipulating?
Yes DH always gives her a belly rub when she does this. I use to but I stopped and just tell her to get up and walk away from her. She then gets up. I have taught her to sit ( with treats) so now when I think she is going down I tell her to sit. Its hard to put a collar and leash on a upside down dog.
Today Bella and I were out in the back yard to go potty. She loves to run and jump but I forgot to take a toy with us so I picked up a stick to throw for her to go get. She cowered and laid down when she saw the stick in my hand. I think she has had some hard times in her short life.

The Father said I've turned my own name back to front and
called you dog, My friend.
-Author Unknown