Would an unpaid vet bill affect my credit score???

Not directly. If the vet sends it to a collection agency, maybe. But if the vet doesn't have your SSN, there's not a lot they can do to your credit. They can sue you in Small Claims Court, though.

I worked with credit for a long time.  Willowy is correct.  If the account is turned over to a collection agency and they are not paid it will negatively affect your credit.  If the collection agency or the vet sue you in small claims court and they win a judgement (whether that judgement is paid or not) it will negatively affect your credit.  In some cases if the vet and/or collection agency is small they may only report to one of the three national credit bureaus, but you can't really count ont that small saving grace. 

If I were you, I would start getting some stuff down on paper, like how much you owe and how much of that the vet is willing to take responsiblity for.  You need to get it on paper and signed in agreement by both of you to protect yourself.  Even if the vet agrees you don't owe them you still need to get this in writing because they could come back later and change their minds.  If you take your cat to a different vet and you feel the original vet is responsible to pay for that bill ... again you need to get the original vet to put it in writing that they will pay it and be responsible for that bill to the second vet otherwise if you go on verbal agreements only you can get into trouble because people can change their minds on what they agree to.

I'd talk to the vet's receptionist and say I need to schedule some time to talk to the vet.  Then when you go have a couple of plans of actions lined up with appropriate written agreements and talk to her and say why you feel she should not only not charge you for her work but also pay for getting your cat back into good health at a second vet's office.  In my experience, even though it's hard, it's better to tackle these things head on then just let them fester.

Unfortunately it is easy to get an SSN.  I work for a company that does insurance billing for fire department ambulance service.  The company has given me access to a commercial data base to find information on patients including SSNs.  I agree that you need to try to work this out with the vet's office.  Many small businesses are very quick to turn accts over to a collection agency.

Thank you everybody for your time and replies to my question...very helpful to say the least,