Not sure if my dozen or so trips on the Dragon qualifies me to give advice, but I'll do my best to give you the benefits of my experience and observations. First, if you can avoid being there on a weekend, by all means arrange your schedule that way. It seems to me that Saturdays and Sundays is the time when you find the largest percentage of crotch rocketeers trying to break the "record" for the 11 mile run or little sports cars such as Miatas coming around curves usually taking their half out of the middle of the road.
During the week, it can still be crowded, but it seems to be a lot more sane. Since you said you ride conservatively, bring that attitude with you. There are some pretty hairy turns of every description on the Dragon, and until you have a feel for the road and your abilities on your bike, take it easy. On some of those hairpins, if the speed limits says 15 mph, do 15mph. That is the best way to avoid having parts of your scoot from hanging on the tree of shame.
Be careful not to oversteer the curves because you will be pretty close to dragging the boards, and if you're not ready for that, you could dig into the asphalt and dump the bike. Best advice is to take your time on the first trip across the Dragon and enjoy the ride and the scenery. Once you get to the end, you'll have a better feel for things and your confidence will be where it will need to be to make another try at it.
The Dragon is NOT a dangerous road, but other vehicles do raise the danger factor more than anything else. One more thing you hopefully won't have to deal with is some 18 wheeler trying to force his rig around some of those turns, and they usually need the entire road to accomplish that feat. Don't be surpirsed coming around a turn too fast and have to lock up the binders because a big rig is in front of you. Above all, enjoy. It is a fantastic ride.