I popped a skin abscess in my chin and some blood came out in the pus. Should I be worried about the infection spreading to my blood?

I've had the abscess for a couple of weeks, which I think developed from an infected cyst. The doctor prescribed me the antibiotic, Clarithromycin, because I'm allergic to penicillin. It didn't work, in fact it got worse. She then prescribed me a further course of the same antibiotic (I guess because there isn't an alternative?) but told me not to take it unless it got really bad, and she booked me for an appointment at the hospital to get it removed. Anyway, when I got home, it looked ready to pop, and I couldn't resist the urge to pop it myself, but some blood came out in the pus and I'm worried about the infection spreading. I feel OK but I did have a slight headache for a while, but that might have been from biting my teeth so hard. I'm still taking the antibiotics, even though I don't think their effective.