Okay, I know what you're talking about now, we don't call them gum boils in the dental world, but that's okay

You definitely need to see your dentist. Yep, your GP might give your ABs, but this will likely pop up again, and you don't want it to be at an inconvenient time (like when days off having a baby, or with a newborn!), and it might well be very painful next time. GPs have NO idea about the follow up treatment that's required for dental abcesses/draining sinuses, so you will need to see your dentist for sure.

By all means get the script off your doctor, but don't put off going to the dentist. Try to do it before Christmas!