I volunteer at a shelter/rescue,for cat TNR (trap/neuter/release). Here in Louisiana, our shelters are busting at the seams & they’re running out of space for animals. Most are desperately seeking fosters to take the animals & care for them in home, until they can be adopted. If you can find a shelter in you area that fosters animals for adoption, they usually pay for all costs involved: food, vet care, etc. You just have to care for the animal & meet potential adopters or attend adoption events.
Maybe this is a long shot, but I came across this article A while back about a woman in Canada in the same spot as you with her grandfather’s cat. She ended up starting a non-profit to home elderly folks’ pets when they pass. She has a facebook page (assume she still does). Although she is in Canada, she went through the same issues as you & can possibly provide some insights.
https://www.westernjournal.com/woman-begins-new-nonprofit-aid-pets-whose-owners-near-end-life/
I hope this gives you some useful information
& wish I could help more. You are so very kind to try to care & rehome the cats, bless you. So many others would just dump the cats somewhere & be done with it. I wish you the best of luck!