Poor little soul. She's not had much of a life until she came to you, has she?
People often don't realise how much time and attention a companion animal needs until it's too late and they've committed (or not, as the case may be!).
So, we know that in her short life, she's had three homes (including yours), cat flu, and a spell in rescue. All of that and she's not a year old yet. Gawd bless her paws!
We also know that when she does poo outside, she likes to leave it exposed, which is potentially a clue that she is sending out scent messages for the benefit of your other cat, and any cats she thinks may happen to come into her environment. This is less surprising given that she's had to potentially compete for any available space.
Her first and second homes may not have had an availability of litter trays, or may have had frightening situations for her to adapt to, such as other cats, animals or small children. She may now feel she has to compete for space, even though she gets on well with your other cat.
We have 3 cats - all rescues. One is a chappie - approaching 5, and then we have the two gurlies, semi-ferals aged 2 and a half. The gurlies gave us some sleepless nights (still do!) and one of them used to pee inappropriately on our furniture and bedding whilst she was settling in with us. She's the most anxious of the two girls, and change affects her most of the three.
We eventually found a solution to the problem, but had to use a number of options in our arsenal, as it were.
I suspect you may have to as well. But you're probably this little lady's only hope if she's to avoid losing yet another home, and facing more upheaval.
I think I would be inclined to increase the number of litter trays (sorry!) by another three to four (yes really!), placed in different areas of the house, including putting one right next to the one she uses for weeing. When she poops on the floor (as she undoubtedly will), I would scoop it up, and pop it in the nearest tray to where it has occurred and leave it there for her to investigate.
The chances are, you won't need to keep them permanently, as she may choose a favourite to use (let's hope so!) and you can gradually remove the ones surplus to requirements.
I know you've already tried a few different litters, but perhaps experiment with something like chick crumb. It's a bit more expensive than a lot of other litters, but it's soft on paws, lasts a long time and it can sometimes win round anxious cats. Might be an idea to put newspaper or puppy pads under the trays while they're being tried out. We have one cat who is very clean but often misjudges the side of the littterbox in her enthusiasm to get close up to the wall, as she feels safer that way. It would make me smile to see her determination to position herself just so, only to find half the pee was in the box and half outside. Still...... we're family.
I'd also try a plug in, such as Pet Remedy or Feliway (preferably in a socket that's lower to the ground, but no matter if all you have is one higher up). They're cheaper to buy from ebay than in pet shops, and you normally get two in a refill pack, but you'd need to buy the heater units first of course first. They're worthwhile though - I can thoroughly recommend them.
I also have a small bottle of feliway spray with a little used from it - if you wanted to pm me with your address, I'd be happy to post it off to you. You can try spraying it around the litter tray and the floor where she is going to the toilet. Pet Remedy do a similar spray. Our gurlies responded better to Pet Remedy than to Feliway, but different cats prefer different solutions.
Hopefully, some of these suggestions might bear fruit. And if they don't, at least you'll have a number of litter trays you can use to grow seedlings or for painting jobs around the house.
(Don't mind my sense of humour - I just think it helps if you can keep smiling while you're dealing with these sorts of issues.)
What are your lovely cats names, by the way? Sorry - I should've asked earlier.