This kind of situation is enormously frustrating and it can drive you up the wall. It's ok--things can get better, but it might require a lot of patience. And time. But things can get better.
However this is an entrenched problem--she's been behaving in a way that doesn't work for all the beings in the house for almost 8 months. This will take time. With a systematic, logical and patient approach, the cat and you will gain your sanity and happiness.
First of all, get some URINE-OFF or another enzymatic cleaner.
2. Locate all problem spots where she goes. a) spray and get rid of all traces b)limit access--move the beds, limit her access by closing doors, etc. c)keep everything super clean--that means, no stray litter outside of the litter tray. Oh by the way, washing is not sufficient to remove scent of cat urine if you didn't catch it while it was wet. So you need to treat it with an enzymatic cleaner. Sorry.
3. It's unclear from your post whether she has been checked at the vet for urinary infections or other biological issues that can create litter tray misbehavior. Try and rule these out first.
4. How many litter trays do you have? Most cats prefer one each. Look at the ways the other cat may be affecting her behavior. Does he have runny stool? Does he go a lot? One of my cats didn't like the tray because the other one had a celiac/bowel problem with constant runny stool.
4b. Look at the cat litter. Is it heavily perfumed? Do you have the crystal litter? Is it clay? dirt? In general, I say keep things very simple. I like wheat litter because it's natural and good for fussy cats but it's a bit pricey. Try and get less perfumey litter. Add a little bit of regular baking soda.
5. What is her relationship with the dogs? Hostile? Can you limit her contact with the dogs for a bit? if she likes the soft dog beds, she probably goes on other fabric as well. Or does she just limit it to the dogs' bed?
You have a lot of things going on--multiple animals, species, etc. It's hard to limit factors and see what exactly is driving her to do what she is doing. Trust me, she wants to make you happy--and with patience, you can get the results that you want.
Just keep in mind the 3 pronged approach--biology (check for infections) cleanliness (litter use and type, as well as no other remaining scents in old litter spots) behavioral psychology (when does she behave in a way you don't like, do other animals make her uncomfortable, etc.) Is she super nervous? A bit strung out for a cat? I really am reluctant to suggest it, but it seems for cats for whom this behavior is caused primarily by psychology (this is a very small minority), valium and prozac can be helpful. Please discuss that with a vet if necessary.