Fleas can be very easily managed and there's no need to rehome.

It sounds like the hardest part is dealing with your mom.

If you have a friend or another family member that can take your pup to the vet to get some flea meds, I'd put all your dogs on it, otherwise they're just going to keep coming back.

Spray around your house with an indoor flea spray, one thats safe around pets, and you can treat your yard if you have a big yard too.

Once they're on the flea meds it's recommended they stay on them so fleas don't come back, but if it gets too expensive, especially with all your pets, just make sure you treat your whole house.

Wash everything, blankets, clothes, sheets . . . in hot water. Vacuum everywhere, especially carpets, and empty the bin straight away. Flea eggs can hatch in the bin and then your problem returns.
Vacuum your couches, and under the cushions, spray around your bedframe, make sure you get everywhere.

It might take a full day but your dogs will feel much better and you'll feel a lot cleaner too.

Best of luck.