I run an animal shelter, mainly for cats, as well as the occasional street dog that I can treat here, with Soi Dog Foundation helping out with the more difficult cases.
Bringing your cats here is not an issue but, in order to organise all requirements - pet travel passports, microchips, other relevant tests, plus certificates of export from the US and subsequent import into Thailand - it would be best to do this via a pet travel company, as it will be fairly time consuming and having someone this end that has the experience, will make it a lot easier and more comfortable for them.
I can DM you relevant details as I have done this in reverse from Thailand to other countries for rescues here.
If flying into Bangkok, there is also pet friendly accommodation available, if needed on arrival.
As you are already well-organised with long-term accommodation, it is just a case of ensuring you have all the correct documentation needed.
Are they indoor or outdoor cats?
Getting a large catio built for outdoor use, which can be dismantled and moved once your permanent housing is ready, may help and afford protection to them. The percentage of strays who have FeLV or FiV is fairly high here.
Last renewal I had (Phuket in April) they gave me a lot of grief over my TM30. I had not gone abroad or even an overnight stay away in a hotel.
As a result, I had to return and get all new documents including a newly completed TM30 form.
I am aware that, at the time, they were targeting to ensure people were staying where they said they were, but I had been in the house 4 years and never had issues previously.
It was very frustrating but I was able to obtain all documentation from my landlords and complete my renewal on the same day.