Thanks, everyone, for your advice. It was a huge relief going to the vet tonight with at least a few clues.
Turns out, she does have a mild UTI, but the vet also thinks that the way it presented, along with her age, also indicated spay incontinence. Good news is, we were able to get a sterile urine sample (thank goodness that Hailey marks like a boy ) and the concentration was good, so there is no indication of kidney problems.
She is on antibiotics for the UTI and Stilboestral (DES) to treat the incontinence---1 per day for 5 days, then wean her down to 2x a week. If she responds well I will try to get her down to 1x a week and see how she goes.