IMHO, try not to draft OF too early unless Vlad is available and never draft a P in the first round, unless the Big Unit is available. And, obviously, draft A-Rod if he's available. That's just my own strategy... many people will take Schilling or Pedro or Sosa or Bonds first round. And I might do so also, in the right situation.
Position scarcity is important, but it's not the only important thing to consider. When looking at position scarcity, look at the next 4 or 5 players at a position. If you would be happy with any of those 5, then don't worry about picking that position just yet. If, however, the top 1 or 2 are heads and shoulders above the rest available, that is the time to draft to position scarcity.
A-Rod is obviously better than all other SS, and he also puts up great numbers in his own right (even as an OF, he'd probably go in the top 3 rounds). Draft him if you can. After that, there are 3 or 4 SS that are all about equal. So, they're not necessary to pick in the first round. Either you have a very high pick, in which case you don't want to waste it on a 2nd-tier player, or else you have a lower pick, in which case there should still be a 2nd-tier SS around by your second pick.
As far as C goes, it's starting to look like a position to just punt this year. If you can get Piazza, I-Rod, or Posada in the first 3 rounds, they may be worthwhile. I picked Piazza in the first round of the mock draft, but I'm not sure that was a wise move. You can't win a draft in the first round, but you can lose one. If Piazza doesn't hit .300 with at least 35-40 HR's, I pretty much wasted a pick, where I could have gotten a Sosa, who's a lock to get at least 50HR's and possibly 70.
At 2B, the other scarce position, things are a little bit muddled. On the one hand, you have Alfonso Soriano and Jeff Kent, who are both great hitters, but one's an aging veteran and the other is still too young to know whether or not he's consistent. If last year was a fluke, Soriano could be a wasted pick too. If he keeps it up, he'll be worth more than A-Rod and Vlad. And then coming up behind them, you have some talented younger players like Junior Spivey and Mark Bellhorn, who could potentially have monster years, but you never really know. And in between them, you have the 2nd tier of 2B, with guys like Ray Durham, Bret Boone, and Roberto Alomar. In fact, it pretty much looks like every 2B this year is a question mark. So, even though there is beginning to be a wealth of talent at 2B, it's still a shallow position because not all of those players will produce what they're capable of (witness Boone and Alomar last year). So draft high or low here based on your own perceptions. I tend to save 2B for last and just pick whoever's available. Looks like it'll be Bellhorn in the mock draft.
3B is the only other position that's close to being scarce. In an NL-only league, 3B is a weak position, but in a mixed league, you've got a little bit more to choose from. Pujols is a great choice, since he plays 3 positions. But don't forget about Aaron Boone. He won't put up numbers like last year, but he could be a steal if other people forget about him.
Remember that 1B is the most stocked position. If you are the 12th person to pick a 1B and you get Derrek Lee or Richie Sexson, you've still got a very good hitter on your team. On the other hand, if everybody else is waiting until the later rounds to pick a 1B, you might be able to snag Giambi or Helton in the 4th or 5th round.
The main thing to try and do when drafting is to zig when everybody else zags. If everybody tries to take a catcher in the early rounds, wait until the end before taking yours. If everybody is waiting on outfielders, you can snag Bonds and/or Sosa very easily.
The other thing to consider is, once you get to the late rounds, don't pick players that start cold and heat up later. Because of your scoring format, it could be very profitable to pick an Eli Marerro or Mark Prior, who is likely to start hot and then cool down. (Granted, Prior will probably be gone before the late rounds.)
Hope I answered some of your questions, and didn't create more. Just my thoughts on how I approach the draft. Everybody does it differently.