I wouldn't shoot steel case shells in an M1 no matter what. Also, unless it was converted to a 7.62x51 by using a "Navy Sleeve" to shorten the chamber, or 3rd party barrel, as the rifle was designed for the 30-06. The '06 is about 1/2" longer than the 7.62x51 NATO. Also, the 308 Winchester is a higher pressure case than the 7.62 nato.
Either way, Butch's Bore Paste on a thick mop and some hand work will polish the chamber
If you use something like Flitz paste be cautious as it more aggressive than bore paste.
Then make sure you remove any trace of the polishing paste from the chamber & barrel, then inspect & a very light coat of oil, then clean & dry the chamber as you don't want oil in the chamber.