Hello, and welcome.
I'm sorry you're experiencing this problem with the multitude of cats and kittens, but thank you for helping them and feeding them, etc. Right now, the most important thing is to remove them from your property, into safe homes and also, to have them all checked by a vet for any health problems, to get vaccinations, and to have the adult cats (and kittens over 6 months) spayed and neutered.
Does your town/city have a TNR (Trap, Neuter, Release) program? If so, you can at least make sure that they can no longer reproduce.
If all the cat rescue organizations and shelters have said they're full, these circumstances are not surprising, especial with winter approaching. Call your local Humane Society, ASPCA, get advice from many local vets--some will keep the cats in roomy cages in the middle of the waiting room and clients will adopt them. The vets will also see to it that the cats are healthy and have received their vaccinations; they might also spay/neuter them to make them more adoptable.
The Humane Society has 'Hav-a-hart' spring-loaded traps for these cats. You put food all the way in the back of the trap and after they walk in, the door snaps shut. You can also cover the the trap all around, with only the front open, with a light material, like a sheet, so they can't see through the back of the trap. Only use these when you are there to supervise and watch the traps. The minute you catch a cat, bring it to a vet with whom you've made a prior arrangement, telling them that you might have to be accommodated as an 'emergency,' due to the situation.
I would not recommend bringing the cats to an area where there are children, assuming families with children will want to adopt a cat. Keep in mind that these are feral cats, not used to living with people, and at the same time, you can't have cats adopted by families your don't know. Any agencies will do home inspections and screen families very carefully. For the same reasons, do NOT advertise 'free cats to a good home,' in the newspaper or Craigslist, etc., because there are many evil people out there who will not treat the cats well!
I would contact Animal Control as a last effort, but make sure none of the cats are put into shelters where they are kept for a certain amount of time and then, if not adopted, put to sleep. Only use 'no-kill' shelters.
Call all cat rescue organizations in your area and even in the next town or city, nearest you.
Ask for help from these big pet advocate organizations mentioned earlier, like the ASPCA and the Humane Society. Also, speak to as many local vets as you can.
If you would like to provide me with your city, state and zipcode, I will try to locate some rescue centers and other agencies that might help.
I wish you much good luck with the kitties and hope you are able to find a happy solution for all involved!
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