They've got plenty of cap space to fix the problems.
They have about $30m right now. A quarterback sucks up close to two-thirds of that, maybe a little less with creative accounting. You could maybe create $12.5m with cuts (Sanu, J.McCourty, Develin if he's done, maybe Burkhead and the like but a replacement is just going to cost the same) and maybe another $10m by restructuring Gilmore and Mason.
So you're looking at maybe $32m in cap space post-quarterback (assuming Brady or a veteran), but in this scenario you need to replace Sanu as well. Figure re-signing D.McCourty, Butler Slater, Ebner, Shelton, Bethel, Karras, and Roberts and you're down to maybe $20m.
You still need to replace Thuney and you probably need a linebacker/edge guy to replace Van Noy and Collins and your receiver corps is Edelman, Harry, and Meyers and you need a tight end too.
So you're going to be able to fill maybe two of those holes. Guard seems like the one you can probably skimp on, especially if they do bring Karras back and expect Andrews back at full strength. Getting a top-level receiver with that much space is probably going to be difficult (a 1st for Beckham and you'll be paying $14.5m of that $20m, though you could restructure some of that away). A couple mid-level guys like Robby Anderson and Demarcus Robinson and a tight end like Henry is probably more realistic. That's a better supporting cast than we saw this year, at least, but you're probably relying on the draft or a lottery ticket in free agency for an edge rusher or linebacker and you're counting on Harry and Meyers to progress (which is fine, you sink or swim with your draft picks to some extent).
