We're all pulling for you, Nicole. Mogs is right. It would be a hard heart that wouldn't dissolve at tears over an owner's distress at potentially losing a beloved pet. Begging is good . If you have children, take them with you. How many vets can turn away a sweet child's tearful face.
If the girl (vet tech, receptionist?) said they MIGHT do a payment plan, they probably will. Of course, any business isn't going to offer payment plans right up front: they're hoping for a flat-out payment that fills their coffers immediately. But the very fact that it was mentioned, was a tip off that they may work with you. If they've done it in the past, they might do it again.
If you have tried Care Credit, it means you were hopeful that they would take you on. And they are nothing more than a structured payment plan. Perhaps eight months is too short a time? The vet might be a little more generous and stretch the payments out over a year. Don't give up. It's not over till it's over.
Also, you could volunteer services..clean kennels and cages, dust and mop the reception room, wash windows and floors for a while, things like that. Perhaps the vet will let you take on a payment plan PLUS do a little work for them on the side. It's worth it to take that kind of leap. Lots of people have fallen on hard times, and these services can be surprisingly sympathetic.
Check out Facebook, Twitter, and Google. Even places like the Salvation Army. Key in "Emergency assistance for families who need help with vet bills." Don't leave a stone unturned, even at this late hour. Check the local SPCA tomorrow, too. More and more, authorities who have the ability to help families consider pets to be important for families' morale, especially if there are children involved.
They will ask some questions to make sure you really need help, but what the heck. It's for your pet, right?
We'll all send our prayers up tonight, hoping for a wonderful outcome. Bless you and your kitty.