Here's something worth thinking about. It is my understanding that if a player like Josh Beckett gets placed on waivers and someone claims him - he has no choice but to go to that team (assuming the Red Sox don't pull him back off of waivers). Of course Beckett would have the option to retire but that would actually be better for Boston as they would be rid of him and his contract but wouldn't face the risk of seeing him succeed somewhere else.
Now I'm assuming that if Beckett gets claimed by another team then he would lose his 10/5 no-trade protection at that time as well. Wouldn't that make him more attractive to a team that likes to take risks? A team like the Oakland A's for example who just lost starter Bartolo Colon to suspension and whose GM likes to make deals. Beckett's peripheral numbers like HR rate are not as bad as his ERA would have you believe. If he did bounce back from a change of scenery then Beckett could also be a good trade chip at the trade deadline. Don't forget that just last year Beckett finished in the top 10 in Cy Young voting. And he's just 32 years old.
Maybe wishful thinking on my part but something to keep in mind nonetheless.
EDIT: Fangraphs asks Should Anyone Trade for Josh Beckett? They think the Nationals would be a good trade partner. I think Beckett would thrive in the NL East (at least for a few months).