Don't worry, they won't ask you to stay. And if it does come to it, they'll do it under sedation at least, or if it's more complex, a general. It all depends on how bad the build up of toxins are. If you caught it early, you may well get away with just tablets.
If they lance, they'll make a fresh incision as the original wound is sealed with muck. Then they'll rince out the area of the abcess, but they won't stitch. The idea is to let the area drain as it heals.
But I'm not expert. Without seeing it could be something completely different... but I've had to deal with a lot of boils, etc.