This thread has some good ideas for live trapping cats.
Now, are they strays or feral? Strays being approachable cats who have been abandoned or let out all the time. Ferals being wild cats who do not trust humans. If strays then you may have a chance of finding homes. If feral the chances are slim. What you then have is called a feral colony. You can still get them spayed and neutered, and we would love you for doing that, but they would not have much chance of being tamed enough to become a pet per se.
You can do a google check to see if there are any low cost spay neuter programs in your area. Or ask your vet, humane society (or your equivalent). They may be able to tell you. A live trap is fairly inexpensive. I got mine for under $100 Cdn. You will also need carriers to transport the cats you trap. It takes a lot of patience, care, and kindness to be able to pull something like this off. Would be nice if you could hook up with a group in your area. They can be a great resource for information, help and support.
Funding for rescues has been and probably always will be woefully deplete. It seems no one wants to take responsibility for the mess we caused. I don't know of any programs available in the States but we do have a few members from there. Maybe one of them have some ideas.
As for cleaning..... it has to be done on a regular basis. Cats smell, pure and simple. Maybe they could use more volunteers to come in and clean/sanitize the litter boxes, clean their cages, whatever. A good cleaner to use that is safe for cats is Nature's Miracle.
This site gives you a recipe for helping with odours.
And lastly, neutering your male...... why would you get your female done and not your male? By doing that you are only taking care of half of the problem. As has been noted, a male cat can spray. When they are defending "their" territory they spray more. If they can smell an unspayed female they spray more. If you get kitty neutered before he reaches sexual maturity (around 6 - 8 months for males) you have less chance of him spraying. Also, an intact male (one that has not been fixed) will roam up to seven miles away from their home looking for females in heat.
I hope this helps. Good luck in your quest. Let us know how it's going. I am sure we can give you advice as you go along. This is a great site for that.