Strategies for rescue dog mouthing at kids

I have a rescue dog who is about a 22 lb terrier mix. The rescue and vet told me she is about 1.5 years old. She has done very well adjusting overall and learned both potty and crate training quickly. She has some behaviors that seem more like a puppy’s, in particularly mouthing and jumping onto people who are sitting on the couch when she gets excited. The biggest issue I’m having with that is that when she’s in play/zoomies mode she will mouth at my kids’ feet, including if they’re walking by her or just dangling their feet off a chair, and sometimes at their arms. She also will mouth at my arms, hands, and feet, but I’m having a better time correcting that because I can respond immediately and shut down the interaction. With my kids she will get a reaction, whether it’s because they’re laughing or upset, particularly my toddler. I’m trying separating her to a more boring location of the house when this starts. So, 1) I’m wondering if that’s the right strategy or if there are other things I should do. 2) I’m wondering how much of this is residual puppy behavior and whether that makes any difference in training (my previous rescues have all been 3+ years old so I don’t know much about puppies). 3) I’m also wondering what training I should be practicing for her to learn to give people personal space and that she can’t just jump all over them when they sit down. I’m worried I might be giving mixed signals bc I have no problem with her on the furniture with us when she’s calm, which is a good amount of the time (she’ll just curl up next to me and sleep). So for example, should I be teaching her to ask permission before jumping on the couch to lie down with me?

Thanks!