Sounds to me like this lot of old military brass is the problem, since you can use once fired brass from your rifle with the full length die with no issues.
I hate to say it, but I would set that lot of brass aside and not modify anything. The US hasn't used .30-06 as a common issue military round in decades. The last I am sure of is the Navy still using M-1's and maybe BAR's in the early seventies. .30 cal Browning's? Probably the 1960's.
Maybe it is age hardened and would be helped by annealing the brass first.
Robert