You're right, you can't deal with it by punishment. You deal with it by alleviating her stress. It sounds like this dog was a kind of extreme case. OTOH a day isn't very long.
Basically with a soft dog that throws a lot of submission signals you need to learn to read the dog's body language and not throw signals yourself that suggest you aren't getting her. E.g. don't stand over her -- greet by kneeling several feet away to let her approach; be very low key, keep your voice, hands, body posture relaxed etc. Your goal is to soothe and relax. Treats/low key praise so she will associate contact with you as a positive thing not something stressful. It's not so much ignore -- ignore is a sort of clumsy approximation of what you need to do -- you need to create a space where the dog can interact with you without feeling stressed (and happy excitement is stress just like scared/anxious excitement).
No doubt she was doing this in an extreme way because being in a new place was undoubtedly very stressful for her, on top of who knows what else, being shipped around the way rescues often are. Therefore what you experienced in 24 hours could have little to do with how she'd be in a week or so. I adopted a "soft" rescue in January and it was weeks before her "real personality" began to bloom. I am now mad for the dog, she is the best dog I could imagine. So I have to say, if you're really feeling so heartbroken you might want to give your rescue another chance and try controlling interactions so that if she does pee it won't be somewhere that's a PIA to clean up.
She might always have a tendency to this but most dogs outgrow it . . . is she a puppy/young dog?
Also has she checked out okay w/ a vet? Always a good idea to have a vet check to rule out physical causes for any unusual urination problem.