Is the drug trade in Morrowind organized to a certain degree? Or is it just something along the lines of a person selling their own moon sugar or skooma?

I believe much of it goes through the Cammona Tong, but probably not all of it. I'd bet the Renrija Krin of Elsweyr has at least some hand in it, if they're not working directly with the Cammona Tong.

Yes, it is quite organized. As has been said the major player behind this is the Cammona Tong and its head, Orvas Dren. They control most illegal trade in Vvardenfell (if not more of Morrowind), including Moon Sugar, Skooma and possibly Dwemer artifacts (that one I'd have to double-check, though at the time of Morrowind they were also working on smuggling the Ash Statues). There are minor dealers in various places outside the Tong, but its only natural that you can't ever get a complete monopoly on something.

I suppose Dren is the big drug lord.
Small groups or individuals would probably get involved to earn quick money too...
unless they decide to try it and got addicted and need to use it for themselves.