This showed up about a week ago. It was really painful to the touch, so I figure it was a spider bite or something. There was also a huge red spot bigger than my fist that surrounded it. Now, it is still tender, but doesn't seem to be going away. Should I just go straight to a dermatologist? I know there are some doctors on board.
It's not a brown recluse, or you would have had a large area of central necrosis by now.
It's not likely lyme disease, because that rarely causes the prominent swelling you have.
It's most likely a bacterial abscess with associated cellulitis. Antibiotics are the treatment of choice, and I don't understand why that's a problem to you.
This will likely resolve without incision and drainage; if it drains or doesn't improve, you need to return for a culture of the exudate.
Your Doc likely made the right diagnosis and treatment. If you want to do that yourself, you'll need to move to Mexico, where you can buy your own antibiotics without a Rx. That's a place where serious consequences occur due to people treating themselves with the wrong medication, wrong dose, wrong duration, for the wrong diagnosis.
God luck to you.