Sounds to me like you need a financial plan.
I think at the first of the year you need to prioritize your spending. If only one of you is working maybe you shouldn't have so many pets. It would kill me also to lose my furbabes but you have a child to think about. If you can't afford to take your cat to the vet, and are scraping $3.00 to pay for gas. You need to do something different. This comes from a person who has done the same thing, trust me. I have scraped money together to the penny to put gas in my car or food on the table. And since noone will lend you any money, you have GOT to figger it out.
You say you have 18 year old college kids as neighbors, go out and ask them to watch Dylan for a couple hours a week while you work. Tell them you will pay them when you get paid. It doesn't have to be for a long time. And you have a computer connection, you find money for that. Believe me, when I was going thru a tight time, that was one of the first things that was gonna go, but I managed to budget it. I know you love your animals, I am certainly not questioning that, but part of the responsibility of taking care of your animals is vet care as they can't take themselves. If Dylan fell and broke his arm, would you wait to get it fixed. I doubt it.
You have the knowledge, splint it yourself. Take a popsicle stick and tape that baby up. A friend of mine had a cat that broke her leg and since we are xray techs we took her pic saw the break and splinted her with a finger splint.
I am certainly not making fun of you, or knocking your way of life, so please don't think that. But it sounds to me that you need some sort of help.