Thank you Karen for all your help! I wanted to post a bit of an update here since it's been a couple of weeks.

I decided to try cleaning the floor first, before going out to buy new food. I have two big bags of cat food to go through yet, so it felt wasteful to get rid of it - especially if it wasn't the food that was the problem. I couldn't quite rule out the floor just yet. The vet had told us that the plastic of her old dishes could be the cause, so why would our plastic-y linoleum/whatever kind of floor be any different?

I've started cleaning the floors every single night with vinegar and water. It's been about 2 weeks I suppose, and it's like magic. Her mouth is 100% fine now. We were away on the weekend so there was about 3 days that I wasn't able to clean the floors for her and I noticed a tiny bit of swelling on the side of her mouth on Monday but it seemed to have disappeared yesterday.

I'm still going to keep an eye on it, but I think the problem may be solved.