[struggles_mightily]

You'd have to give up a pick, of course (presuming that the player involved was a Type A or B free agent and had been offered arbitration by his former club). So it'd be up to you to decide whether the value of the prospect you were receiving would be greater than a player you could potentially take in the amateur draft at a particular point.

In reality, doesn't MLB have a hard upper limit on the amount of Type A free agents that one club can sign in a year?

Those two rules together might prevent this from being a viable strategy, as you reap larger rewards the more times you do this in one offseason (since the picks you have to surrender are of less and less value with each free agent signing, but the calibre of prospect received presumably remains constant).

EDIT: All in all, I think that this probably is an effective strategy in most cases -- particularly if there are Type C (or whatever? "unranked"?) free agents available that a team in your league covets but can't afford. If signing a Type A or B player, then it's still a good strategy as long as the value of the prospect you get back exceeds the value of the pick you lose (or the total value of all the prospects you get back exceeds the total value of all the picks you surrender, if making multiple deals). I do think that "moral"/ethical question are, as you imply, quite important in explaining why baseball doesn't witness "sign and trade" moves.