I think the lady at the doctor's office mislead you. I was in a similar situation last year, where I was between jobs, and needed to see the doctor, but I hadn't paid my Cobra premium yet. I just went to the doctor, telling the office I still had the same insurance (they already had my card on file). As long as you pay your premium before the claim is received by the insurance company, there shouldn't be a problem. And if the claim is reviewed before you pay the premium, it will just get rejected and will have to be re-submitted.

Either way, you shouldn't have to pay up front for the entire cost of the visit. And there is a grace period that you have to pay your premium (I can't remember exactly, but the 60 days sounds correct).