My signture is going to make this appear rather strange I suppose, but it's amazing the things we pick up over the years. First, are you still waxing? if so, please stop immediately. As unusual as this is going to sound, you may find not only relief, but an elimination of the soreness and swelling.
Fill either an atomiser or clean new spray bottle with 2 parts white vinegar to 2 parts water. It would be better to use water that has either been filtered or previously boiled in the kettle. Spray the infected area, allowing it to naturally dry. Continue to do so approximately six-eight times throughout the day. You should find that in minimum cases it clears up overnight. In moderate cases, it should clear in a couple of days.
If you dispense with the waxing and revert to a razor, you'll find that by spraying the shaved area immediately after shaving, with a men's aerosol deodorant, you will prevent any redness from occuring.
In my 'much' earlier life I was a photographer and picked up this little household hint from one of our cosmeticians.
I would continue to take the antibiotics - hopefully it's something along the lines of metronidazole or tetracycline which work on the dermis or tissues, rather than cephalexins. The only reason I comment on this is that I believe it may work slightly quicker as well as preserving the more aggresive antibiotics for other emergencies.
I wish you every success
Fr Bill