He said at a Con that he doesn't smoke pot. He grew up in a house where there was pot, and he said he was very anti-pot for a long time. He has smoked it, but he said he is "the worst stoner ever." So unless you count nicotine as a drug, I doubt he's a drug addict. The man smokes like a fiend.
I personally think it's ADHD. Watch him at any Con, it's pretty obvious. He plays with toys, draws, takes pictures... He did one Con with Bernthal and he was playing with his mic (taking it apart, blowing into it, etc.) for almost the entire thing. One of the moderators told him to pay attention during someone's question, and he answered in a way that proved that he WAS paying attention. The microphone was just very interesting as well ;) Plus, he's said he's very shy and socially awkward. So no, not drugs... it's just part of the beauty that is Reedus.
Norman doesn't do drugs, pot included... he's had family with those issues and seen what comes from it and if you ever meet him at a con, probably best not to ask if he's some stoner or something either. ...he also has ADD which explain the mannerisms....it scares the fuck out of me that people equate ADD mannerisms with drug use now too...jesus christ.