Our plan though is to live in a place for 1-3 months, then move to the next country, live there for a bit, and then move on, and so on and so forth. We plan to travel the world for years.
Awesome, that's my FI plan too! I'll probably try to get one semester, one class teaching gigs just for fun (we'll see if that part works out, but at least it really just would be for fun and a way to meet more local people.)
On to the cats, here is an idea if you want to try it: My mom used to bring in tame strays quite a bit which she would place and homes and help people she knows rehome their pets. She always used the same strategy that involved placing a "free to good home" ad with a description of the pet and the key phrase "will deliver."
By delivering she got to see not just who was adopting but also where the pet was going. She wanted to avoid pet hoarders, for example. I can think of one time when I was with her and she just kept on driving because of the really horrible condition of the home's exterior. That may seem snooty, but she wanted someone who could and would take care of the animal. Several times this led to a person with mobility issues who wasn't driving. Sort of the senior home idea someone else posted, although the people weren't always seniors. As long as they seemed to have a good system for getting supplies and self care in place (and nothing beats a home visit for determining that) she was fine with those folks. You can pretty much tell watching someone interact with an animal if it has a good chance of being a good fit. Harder with cats, who tend to freak out when traveling, but you can observe the person to see if they are understanding of the pets behavior.
Also, and this is as key as the "will deliver" part, she always left her contact information when delivering the pet and asked the people to call her in the next couple of months if there were any problems at all and she would come pick the pet back up. One dog bounced back once ("too rowdy") and one cat came back from two different homes ("scratched the children"). In both of those cases she was able to find a better fit for the pet. Most of the time, though, she felt really good about the home where the pet was going when she delivered it and she never got a retrieval call. A couple of years ago I was able to home a cockatiel we found outside with this method and I was extremely happy with where the bird went (a couple who had one other bird who was obviously being babied.)
The drawback of my mom's method is that she had to deal with a lot of bozos who contacted her (and I had to deal with some crazy bird people when I placed the bird ad.) The vet where I took the bird to be checked the day I found it said to say it was found but not describe what the bird looked like in the ad, and this seemed counter-intuitive but turned out to be good advice since I was at first trying to find the original owner of a lost bird, not someone who just wanted a free bird to sell as apparently some birds are quite expensive. Mom wouldn't even consider going somewhere if the person didn't ask what the pet looked like. She was also skeptical if she had more than one pet and people wanted both but didn't ask for a description of them, as in the situation where she had two cats who had been living in her work warehouse. That is what makes your situation even more difficult because not many people want two new cats at once, although I adopted two adult cats at once myself so there's your counterpoint :-)
Good luck with finding them a home, ars! It is a very slow process, but it sounds like you have plenty of time and if you are persistent you will find a situation that you are very happy with. I had a friend find a home for her cat when they were moving to Hawaii because of the quarantine issue. Another friend who moved to Europe had to quarantine her cat for some ridiculous time like 3 months, and she said it was horrible for the cat. I think that you are being very responsible with the way you are going about this, so just hang in there and you'll find an awesome set up for the cats!