Would the Red Sox even be allowed to trade Drew? I can't find it in the rule book, but I read on fangraphs that sign-and-trades aren't allowed. I also read that Tim Dierkes, from MLBTR, said that MLB would view it as collusion.
Looking back at 2012, you can see the impact the QO had on contracts. There were players that were easy QO offers, because they were going to get more than 1/13, but Drew isn't one of those players. He fits more in the category of Bourn, Lohse, Soriano, and LaRoche. They all signed below expectations, and they all had a very low number of interested suitors. Actually, I think they only had one interested suitor for each of them. Lohse had the Brewers. LaRoche had no other choice but the Nationals. The Nationals were Soriano's only interested team. The Mets were interested in Bourn, but they balked at surrendering the 11th pick. Compare them to the players without QO's attached.