Don’t get rid of your kitty, please. Just get a cat feeder that holds enough dry food for a few days and keep it full, and one for water. That way if anything happens to you and you have to go into the hospital your kitty will be okay.
All elderly people who live alone should make some sort of arrangement with a neighbor to check up on them or call a family member if they do not see them for some agreed upon period of time. (And feed kitty.)
My cat is used to being on her own for 5-10 days.
I leave food and water [AKA: open toilets and filled sinks] knowing that water is more important than food.