Catrin, if you are allergic to any drugs, that can definitely influence the choice of antibiotic prescribed. I cannot take whatever the "gold standard" drug of choice for cat bites is because I have a sensitivity to penicillin, which the first choice drug contains. So I'm on an also-ran, which is okay by me since it seems to have done the trick! (For the record, the bite isn't on a finger, but on the back of my hand, with the biggest wound between my thumb and wrist, where the nerve that goes up the inside of my thumb is. He literally not only bit the hand that feeds him, he also literally hit a nerve. That's why I have a "loss of sensation" in my thumb.)

Have you felt the wounded area to see if it feels squishier than it usually feels? If you've got stuff draining, I'm a little surprised you don't have any swelling, but maybe it just isn't very obvious and you have a pool of .... something... under your skin that is very diffuse. (Isn't that a charming picture I've painted for you, of a diffuse pool under your skin? Ick. I'm sorry!!)

Good luck at the doctor's!