Is my cat sick? She is acting weird and going outside the litterbox?

Sorry for the long question, there is a lot to tell-
I have a female cat named Bubbles who is one year old and has a very sensitive stomach. About a month ago me and my boyfriend were giving her two different types a food a day for over a week (unknowingly), and she started to get diarrhea. Around this same time we got a new kitten. She had so much diarrhea that her butt was raw and she was pooping blood. I took her to the vet and she had a fever but otherwise seemed ok. (She was also freaking out at the vet so the fever wasn't abnormal). I got antibiotics and her diarrhea got better and her butt got much better as well. Her poop is still runny, but she isn't in pain anymore and she is more playful. The only thing is I recently noticed she is going poop and pee outside the litter box, as well as inside of it. I have 4 cats and two litter boxes, and it has never been a problem with her before, although one of my cats died and a new one is in the house now so it might be different for her. In addition to this she is acting a little strange. She will play if you engage her and she will play by herself as well but she doesn't seem 100%. She is eating and drinking normally but like I said something isn't right. Her health problems and weird behavior started when I got a new kitten, she doesn't mind the new kitty but isn't overly thrilled about it either and I can't tell if that has something to do with it. I also wasn't sure if that had anything to do with the going outside the litter. I have had multiple cats who have gotten feline leukemia and had other health issues and I have spent thousands of dollars at the vet. I really don't want to spend anymore unless I have too. What could be wrong with her?

I'm afraid we can't really diagnose a health issue over the internet. 

One thing I will say, though, is that you probably need more litterboxes.  The rule of thumb is one box PER cat,  plus one more.  Especially if you've got a cat with runny poop, I'm guessing when she needs to go, she needs go, and if someone else is in the box, or there isn't a box nearby, she's got a problem.  (think of yourself when you've really got to go and can't access a bathroom for whatever reason )  Now I know it isn't always easy fining places for litter boxes, but I'd say you really need to try.  Heck, I've got one in my master bathroom bathtub . 

As far as Bubbles acting strange, has she been tested for FIV/FeLV, etc, and been vaccinated?  And is she back on her correct food?  The runny stool is indicative that something is still not right in her digestive tract, so perhaps she still doesn't feel well in that aspect, even though her behind is no longer raw.  Has she been diagnosed with anything specific as far as her sensitive stomach?  Like IBD or just "sensitive stomach"  Do you give her probiotics? 

She may still be mourning the loss of your other cat ,  and/or she may not like the new interloper yet.  A month really isn't all that long to completely accept a new kitten, especially for a female.  Since her strange behavior started when the new kitten arrived, I would guess it's all related to this, and she may be "sulking" (or mad).  Did you go thru the proper integration procedures?  Maybe try just brushing the new kitty a couple of strokes along one side, then without cleaning the brush, stroke Bubbles along her side,then go back to the new kitty and brush her again.  This transfers the scent between cats and helps.  I do this with my cats even now when I sense a little stress between them.  In your case, since one is new to the "hood", you could do it several times a day between the new kitty and all your other cats, only cleaning out the brush between the resident cats.  New kitty, resident cat, new kitty, clean brush.  New kitty, different resident cat, new kitty, clean brush.  repeat as appropriate.

She has been vaccinated and tested and she came back negative for FeLV. She never got a specific diagnoses, I was just told "sensitive stomach" was the issue. Yesterday after I posted the question, I locked her and the new cat away from each other and gave Bubbles her own new litter box. Her behavior increased dramatically and she seems much happier. I am going through the proper integration process today.

Sounds like a good plan