Heading into the beginning of free agency, let's say you are targeting a type B (or type none) FA that you want. You don't expect a bidding war for this guy. Do you offer him what he's asking for and be done with it? Or, do you roll the dice and wait for the offer to come down first? Let's assume he's asking for less than what he made last season, but he's also regressed some as well. Just looking for any best practices here (i.e. safe ways to tell if the guy is getting bids).

Depends how bad you want him. This is where being in the league for a while helps. You can gauge who is going to be bid heavily on and who is going to be passed over. I'll usually wait to be honest.

If I want him I pay him if I feel he'll be an impact player and fill a void. I may be creative with the breakdown of the contract, however. For lesser roles, I'll target guys with similar skills. If the better ones start getting scooped up early, I'll jump in so I don't get shut out. If they hang around and the demands start dropping, I'll strike at a point I'm comfortable with.

Agreed with hockey1984.
I'm in two worlds; they behave completely differently in free agency. First one, about ten of the owners are always somewhere in a tanking, uh, "rebuilding" cycle. In that world there's usually low interest in boring older players. lots left to choose from during spring training.
In the other world, heavy competition for FAs. In such cases Type Bs and non-comp players are often more heavily recruited than As. I just signed a Type A, but was eyeing a lower-cost vet as my "Plan B", kind of player who'd give you 70-80% of the Type A. Someone else signed my Plan B guy on the first day.

IF I really want to see if I can get a player on the cheap, that means waiting for his demands to start coming down. Far as I know, that'll only happen if, after the first full day, no one is bidding on him. A player like that, I'll Target him and track his demands on the page. If you see it start to come down, you know no one's bidding on him yet. But it does leave open the door for someone else to pay full price.