I had ocd as a teen (from about 13 onwards) , it didn't fully leave me until 21ish. But when I left school at 18 it got miles better. And then slowly improved after that until it went completely.
I think removing the main stressor is the most important thing. But: that may not be the cat. I Germs might be your trigger (it was mine too) ...but even that isn't necessarily the stressor.
I look back and realise I was ocd because I didn't have control elsewhere. I was horribly unhappy and trapped and bullied at school. That was my main stressor.
If there is something making you feel really trapped or unhappy then maybe look at addressing that. That being said, this might be harder to do while the cat is about, exasperating things.
Perhaps you are just naturally stressy and the therapy helps with that but just thought it was worth suggesting that if there is a main stressor - do whatever you can to change it. Eg: if you dont like your job and it makes you miserable, change jobs.
I understand though that for some, ocd is lifelong and a condition to be managed, not overcome. But don't lose heart, if I can do it, so can anyone. Don't beat yourself up too much if you ultimately choose to give up the cat. Your mental health is what is most important.