For the first several rounds (differing opinions on the exact number but I say seven in a ten team league) I would not pay attention to position scarcity and draft the best available player instead.
If you're smart, you will go into the draft with a "cheat sheet" that ranks all the players from best to worst. If you don't have the time/energy to make one yourself, there are several credible ones out there offered by ESPN, fantasy mags, and other sources. DO NOT go by Yahoo rankings.
Also, "mock drafts" which you will see mentioned around these parts will give you a good idea of where the players are going in other drafts.
After round seven or so, start customizing your draft strategy a little depending on how it's gone so far. If there's a non-filled position with slim pickings left, this is the time to get one of the few producers remaining. But don't panic and sell yourself short. If you're fairly confident you can pick up a solid player at that position later in the draft, wait for him. For example, 3B Joe Crede should be available well after the eigth round comes and go and projects to be a decent man for the spot if you're desperate late in the draft.
I can't re-iterate the importance of a cheat sheet. With one, it will be much easier to make positional decisions when the time comes. Without one, you will wind up with really weak positions or waste draft picks on mediocre position players and miss out on a lot of talent - both are bad situations to be in to start a new season. I hope I didn't teach down to you or anything, I'm not sure how much fantasy experience you have.
That's awesome that you go to meet Bench and Rose. I got to meet David Justice and Ricky Ledee during their playing days.