I will admit this is one of the most frustrating questions I get in practice. It is hard for the owner and the vet and sometimes we cannot find a solution. It is also the most common reason for otherwise healthy cats to be euthanized in veterinary medicine. . . I certainly don't recommend that and always try to talk my clients out of it. But my point is that it can be extremely frustrating. I have dealt with it for 12 years with my own cat so I understand your pain!
First we need to rule out physical causes of inappropriate urinating like FUS or lower urinary tract disease. Your vet can see signs of that often on routine urinalysis (such as blood in the urine). I would certainly rule out UTI in the new female since she has never done this before. That is a simple fix in most cases.
If no physical cause can be found then we are left with behavioral or emotional/mental issues. As a general rule of thumb cat behaviorists will tell you there needs to be one litter box per cat plus 1. So technically they would recommend 4 boxes for your home. It is important to make sure the box is cleaned regularly every day. Clumping, low odor litter is usually tolerated best, ***** ***** cases sand works well. The boxes should be in a location with some privacy but easy to get to. They should be away from doors or windows, and in low traffic areas. In some cases the location of the box is the problem. Look around and see what noises or objects are near by and can they cause the cat to avoid the box. If the cat is more nervous and tends to hide during the day, the box shouldn't be hard for her to get to or in an area that makes her have to come out in the open if she doesn't want to. Also, most commercial boxes are too small. You want something big enough for them to stand in comfortably, get out of easily, and be able to turn around. A tupperware container sometimes is better. You can cut a hole in it too if you have a taller one which also helps with cats that tend to spray next to the box while standing in it.
Sometimes the issue is stress induced and can be cyclical. Sometimes it is an issue between cats within the household. They have been together for years, but that doesn't mean they love each other. Cats tend to be solitary animals in general, rather than pack animals like dogs. If they lay together and groom each other then you are probably ok, but if they tend to avoid each other, even if not fighting, there may be an unspoken dislike between two of them. This does add emotional distress sometimes. Sometimes girls just don't get along :) Sometimes the issue is with someone outside the home, a neighborhood cat or possibly wild animal hanging around causing the house cat distress.
Then there is the issue at home. Sometimes they are upset with you or changes in the home, new people visiting, changes in work schedule, people leaving, etc.
In some cases I can adjust the stressor, change box locations, litter types, and solve the problem. Feliway diffusers help in some cases. They provide a natural pheromone to help stabilize their moods when stressors or competition are the issue. This is probably similar to the mood collar. I have used anxiety meds as well in these cats such as prozac in cases that I just can't solve or where the stressor can't be avoided.
Your vet may have discussed some of these things with you already but I hope it gives a few more ideas. Like I said, it can be very frustrating and stressful. I used to have aluminum foil lining my bedroom walls to keep my cat from spraying, and ultimately have ripped out the carpet in several houses from her. She now lives on the patio and we are both happier. :) Let me know if you have any other questions or ideas.