I got in touch with my program and picked up a claims form for a walk-in office. The office is actually right across my school, on Rue d'Assas. It was super simple and very quick. I was in and out within 45 minutes. Also, the doctor who saw me was really nice. She took one look at me and was like yup, you've got pink eye (but in French). She asked me all about my symptoms and did a short family history, which I have to admit makes me feel pretty good about my fluency in French. All in all, the visit cost me only 23 euros.
The doctor gave me a prescription for 4 different medicines - two for my eyes, one for my nose, and an anti-fever medicine just in case. I was so nervous that I was going to have to pay over 50 euros for this stuff, but it actually came out to 11.50. I'm also pretty positive that French medicine is made with witchcraft, because I took my first dose at 1pm and I already feel way better, but, ya know, that nap helped too.