Gary , appreciate the issue , I love cats too , but you need to get them placed somewhere there , de sexed or not the last thing this country needs is 13 more .
Plus where would you possibly live ? You'd be in breach virtually everywhere .
This would be a very expensive option when you considered all charges and with a foreign card , 7/ 11 - 15 baht ( I know this is not what the OP wants ) .
I had to go to Australia and ended up staying 8 months , what I did was renew the Visa early with reentry permit , and pretty much left immediately , then got back in with 3 odd months to spare and renewed .