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Usually I only bed the receiver and recoil lug. I have a Remington 700 on which the barrel was bedded to the end of the forearm and it does give the vertical stringing you refer to as the barrel heats up.
Savage actions don't bed as well as Remington actions... you can hang all the barrel on a 700 if the action is done right.
I would suggest you bed about 3 inches of the barrel and the action all at the same time with no stress... build the forend up with masking tape to hold the barrel and action centered and level while the bedding cures. One action screw in very lightly just to hold it square. Drill out screw clearance after.
In addition to bedding, check that the barrel is free floated well enough so the barrel is not contacting the stock (slide a piece of paper between the barrel and stock, should move easy with no contact).