I agree that it sounds like the action is binding as accuracy drops when the screws are tight.

You said you ensured the screws were not too long and the mag box isnt binding. With those things out of the way my first move would be to skim bed the action to make sure all is sitting right before anything else.

I'm betting groups improve. If still not where you want try adding some pieces of cut up credit card under the barrel at the forend. I would start with 3-4 pieces to ensure some upward pressure and shoot a group. If things tighten up a little add another piece or two and see what happens.

My 300 WSM shot better then yours averaging consistent 1 1/2" 5 shot groups out of the box. Mine had no mag box binding and the screws were fine. I started with 4 pieces of credit card under mine per Boxers recommendation and it shrank groups markedly so I damned up the forearm with modeling clay and poured an equivalent pressure pad. Mine now shoots pretty much everything under 1" and its favored 150 TTSX into 3/4" at 100 yds more often then not.

At this point the limiting factor is likely me shooting a light rifle but I'm perfectly content with it.