Also, why are you getting rid of the cat? A flea dip, some flea medication, and a flea collar worn for about 6 weeks will ensure that kitty is free of them.
We have 2 small dogs, they get a tab flea med monthly. They think it’s a treat. Same goes for the heart worm med.
Dawn original dish soap smother’s fleas on critters Vet recommended for our small female Morkie. Who seldom needs bathing, they don’t develop the smell other dogs do. Peekapoo smells in 3 days after a bath.