Here is the link for the requirements on bringing a cat into the US.
CDC - Bringing an Animal into U.S.: Cats - Animal Importation
Cats and Dogs
Importation of cats and dogs is regulated by the
Centers for Disease Control and Prevention (CDC).
Cats
All domestic cats must be free of evidence of disease communicable to humans when examined at the port of entry. If the animal is not in apparent good health, further examination by a licensed veterinarian may be required at the owner's expense. There is no requirement for a rabies certificate but most airlines require a veterinarian’s health certificate. Cats arriving in Hawaii or Guam, both of which are free of rabies, are subject to that state's/ territory's quarantine requirements.
So you need to know what the State you are coming into- from Canada- says regarding certificates.
I am sorry that you are losing your dog. It is never easy. I have had to do that five times and it is heartbreaking. I've held each and every one of them, as the Vet administered the drug, and felt them leave this world. It never gets easier but I always knew that they were going to a better place and they would no longer be in pain.
I did find that it was easier for me if I had another one of my dogs with me when I went home afterwards. Don't know if your cats like to travel in cars though. If they don't, it likely isn't the way to go for you............
Take care,
Sheila