Hello,
I am so sorry to hear about your loss.
Cat Chat’s rehoming advice page may help, which also includes advice on ways to avoid having to rehome:
https://www.catchat.org/index.php/rehome-pet-cat
As you have discovered many shelters are full, but it is worthwhile contacting ALL the shelters in your area. Find your local rescue & rehoming organisations here:
https://www.catchat.org/shelters
I wouldn't really advise trying to rehome these cats privately, unless it's to someone you already know. Usually safer to enlist the help of a rehoming organisation - some will help with "direct" rehoming
https://www.catchat.org/index.php/direct-rehoming-cat. If you are on a tight timescale, you will need to let rescue organisations know that, so that these cats can be prioritised.
You could also email Cat Chat’s Help Desk for further advice: cat.advice (at) catchat.org
Hope that helps.