Move the players we can replace immediately, which have value enough to help re-stock the upper minors.
Those players are Salty, Aviles, and Ross. Maybe throw in one of our middle relievers like Albers or Miller as well if you can get someone to overpay (unlikely). And when they're gone, you call up Kalish, Iglesias and Lavarnway and see if you can't luck out, have those guys produce at about the level of their replacements collectively, and use the trade return return to stay competitive.
No Beckett. Sorry. You're not going to get an ROI worth your trouble there. Teams don't take on the players you WANT to lose without you making it worth their while, and no.
No Lester. I don't sell low on LHSP that talented.
If things take a turn for the worse between now and 7/31 I put Papi on the block too, just to see what I could get for him. I doubt he'd be moved, but you never know. I didn't think Youk would be traded.