My abscess was drained under local anesthesia - my choice, since I was in the office and the CR surgeon offered to do it in his office then and there, under local, or schedule an O/P surgery and put me under. I wanted it over with so chose the local.
The shots hurt a lot, I won't lie (they were just outside the vag. opening, which I realize is not the same location as yours, but still ...
Relief was indeed immediate.
Sorry you're going through this - good luck.