Maybe redundant but a few tips: increase her exercise very gradually. With an older dog I wold be careful with sharp turns and sprinting after balls (avoid the Chuck It).
Don't feel sorry for her history. She lives in the present. Susan Clothier had a great quote "A new dog had three questions: 1. Who is in charge? 2.What are the rules? 3.Where do I fit in?" You have to answer them the moment they set foot in your life or they will fill in the answers. It gives them security.
I have seen how my teacher dog demanded this from every new foster dog that was brought in. I took the new dogs about a month to fully fit in my own "pack".
I enjoy following her story.
Congrats!! She's a pretty girl! Sounds like she's settling in, that was long process!! After she settles in more she may start to play ball, or may like other games! Our dogs liked soccer balls! Cody liked to play with toys, fetch, but when he was done, he was done! Clipper on the other hand acted like he'd have played all day and would keep putting the toy on your hands after you were done!
She's a gorgeous dog- as pretty as they get IMHO. It's great you were able to rescue her and give her a great home. I wouldn't stress about the weight too much. A new routine, food and happy life will probably dictate that : ))
Do you have any info of her previous life? She may have been left alone for long periods and never really learned how to play. There is a video on Youtube about the border wall dogs from the Berlin wall. The ones that weren't good for bite work etc.., were essentially tied on short runs on less threatened areas as alert dogs. They literally had no life - just eat, bark, sleep. Some of the people who ended up adopting those dogs had wonderful pets, but they didn't know how to play with people or other dogs. The good news is they learned pretty quickly. Sorry so long.
Luc spent ~3, 3.5 years in a no-kill shelter and took a long time to learn to play - try out different things to see if there's something she likes - Agis likes to chase balls, but Neb could care less (but bring out a leather rag to tug with, oh boy!).
She's beautiful though, and it sounds like she's settling well.