I'd say my encounter with the series was pretty similar, although I'm not sure when it aired on Adult Swim or when I caught it, but I think 2004 or 05. I don't think I ever caught all 50 episodes though. I remember it being such a tease at the time when Conan returned to being "Jimmy" for the first time though.
It did have a really crappy time-slot though, which was quite frustrating. (I think 1am EST). I remember buying a bunch of cheap blank tapes down at the dollar store to record it after almost immediately falling in love with it after seeing the first few episodes (Them TDKs!).
Unfortunately, I moved out soon after, and my father relocated, and I have no clue where the tapes are. It's a shame too, not that I couldn't rewatch those episodes, but seeing all the commercials, and those special bumpers before each episode was pretty cool! I haven't really "watched" Adult Swim in years, last time I flipped it on I was really disappointed in what has become of it though. Sometimes I like to just wallow in my nostalgia and let it consume me.
I suppose I take things a bit for granted now, given it's been a whole decade later, but I'm still in love with the series despite some of its flaws and some of the recent story arcs.
Sorry for rambling!