Well, you could have a short chamber since the 30-06 no go guage allows the bolt to close but with difficulty and this can be rectified easily by a gunsmith by cutting the chamber properly; but I would like to ask if you dropped the no go guage into the chamber and then tried closing the bolt or if you chambered the guage from the magazine? Same question for the empty shell casing. I doubt the barrel was set back and the chamber cut for 308 as it would not allow the bolt to close on the 06 no go guage if it had been.
Make certain the chambver is nice and dry and you should be able to pull the bolt out and drop a dummy round right down into the chamber and hear it go 'clunk', sorta, maybe sort of 'snick', but drop right in. You should then be able to take the rifle and either shake the round out of the chamber or fling it across the room - this indicates you probably have a properly cut chamber and how 'generous' it may be since it is mil-spec.
Is the serial number on the bolt matching the serial number on the rifle - if not it could be a poor fit and may need some gunsmithing. Have you looked into the chamber to see if there is any roughness or burrs plainly evident or that something is plainly out of whack?? Let's keep trying, those are nice rifles. HTH.