There's really nothing wrong with apologizing; everyone does it, even to cracks in the sidewalk when they trip :-) And how about when someone bumps into you, both people apologize, doesn't matter who was the bumper and who the bumpee.

I would just accept that I apologize a lot, it's just a quirk/habit/symptom that will probably dry out and drop off when I feel better and my mind has more things to occupy itself with than my mistakes? The only time I have trouble with other people apologizing is if they apologize for the same thing too many times, like I didn't "accept" their first, "I'm sorry". If you do multiple texting to your friends about the same incident, then I'd worry. Cheer up, luv!