I do not have any advice about rehoming in your area, but regarding your cat:
when she pees (inside or outside the box), is it a very small amount?
or, when she pees in the box is it completely soaked (i.e. you've had to change the litter more often than you used to)?
though she has been given a clean bill of health, many cats get PUPD (drinking a lot, peeing a lot) before being diagnosed with kidney disease. Reduced kidney function doesn't show on lab values until 2/3-3/4 of the functional kidney is lost. If that is the case, it is possible that tests come back normal but she may eventually be diagnosed. I think it's unlikely she has this though, because she is so young (they're usually diagnosed around 5-8 years of age), but it's a possibility.
Also, sometimes (though not commonly) spaying can result in "spay incontinence," where the bladder is more sensitive to distension after spaying (because of the removal of estrogen). If she pees very little at a time, this is another possibility. Unfortunately, this type of thing is diagnosed usually as the last thing left in the list, since there's no real way to test for this. In this case, it's more about the cat feeling urgency to go frequently rather than learned ok and not-ok areas to pee.
Also, it might be worth considering trying different products for urine odor removal. You might not detect an odor, but it may well still be there to the cat. I use Nature's Miracle, which works well for me, but there are a lot of other products out there. You can also check for urine residue with a UV light (it will glow). You can also spray something citrus scented on that area (there are commercial sprays, but it can be as simple as orange oil + water in a spray bottle). If it's possible, you can place double-sided tape (or doubled back packing tape) where she tends to pee; though some cats avoid vinyl, that sort of thing has never worked for me... but they really don't like stickiness. Supposedly they also don't like foil, but I've never tried that one. For your couch, they make rubber mats with soft spikes that you can leave on the cushions or armrests that cats will avoid (and you can just lift one up to sit down). I usually see them at pet supply stores, but they're probably also online.
And, though this might be something you've already done, it might be worthwhile trying out different litters. If it really is just a behavioral issue, then it might be a matter of finding a substrate that she really prefers to carpet/etc. I have noticed that the litters that are really great at covering up odors (pine, large granule clay, and sometimes scented) are less preferred, maybe because of their strong odor, or maybe because they mask odors so well that the cat doesn't associate it with where s/he goes to the bathroom (and then instead pees on whatever smells like urine, like your couch). It looks like you already spent a lot going to the vet, so maybe trying different litters won't hurt so much.
And finally, her peeing location might just be a matter of... location. If you have enough litter boxes, could you move them right where she likes to pee? If it's an inconvenient location (like on top of a couch, haha), you can place it as near as possible, and/or gradually move it from the inconvenient location to a more desired one as she learns to pee in it. If you have a rug (bathmat, kitchen rug, etc) that is so saturated with urine that you're about to throw it out, you can place it (or part of it) in a litter box, and add litter as she learns to use it. Also, I'm not sure how big your place is, but with incontinence problems it might be good to have at least one litter box per room, and to clean them out at least once per day (assuming she ever pees in them...). Sometimes cats with multiple boxes pick some to have for just peeing, and some for just pooping. It's possible she's using the two you have for #2 and considers the rest of your place her #1 box. If you get a new litter box, and happen to have something saturated with pee/there's a puddle somewhere, you can try getting some of the urine in the new litter/box to "mark" it for her.
Anyway, I know many of those things might just be repeats, but may as well try everything, right? It seems like you've been pretty diligent with everything thus far and you want to give your cat the best chance possible. It might mean hanging in there a little longer, but maybe you'll find something that works for you and your cat.

Has the vet ruled out diabetes, UTIs, kidney disease, or other illness? Inappropriate soiling is often a cat's way of trying to tell you that something is wrong physically. Bloodwork might be a good place to start.