If it hurts when she goes, she will look for something soft like a rug. Also, some cats will avoid the litter box due to the volume of urine when their diabetes isn't really regulated yet. (Apparently, it gets on their paws and they don't like it.) Tisha has done a lot of this over the past few years. We have tried lots of things, but have found for right now that having puppy pads where she tends to go is helping. She doesn't usually go outside of the puppy pads or litter box now. They also help with her annoying habit of using her paw to scoop litter onto the floor, then pee on that. (Clumping litter stuck to the floor is really hard to clean up.) She also really likes to use the Breeze litter box (non-absorbent litter, drains to a cat pad in a tray underneath) for urinating. She still thinks stools belong in the regular litter.
Hope that helps! If you continue to have issues, other things the vet has had us try:
More litterboxes
new litterbox (they can absorb smells)
bigger litterbox or litterbox with higher sides
different litter - softer grain, clumping vs. nonclumping, scented vs. unscented
confining to a small space with just food, water and litterbox to retrain (we couldn't do that because Tisha completing freaks out in a confined space)
I'm glad to hear that Claudia is feeling better. I hope she continues to improve!