Cartridges do not have headspace.
Not quite true. The headspace on the 308 round is measured from the center of the shoulder to the back of the rim. That's what case gages measure.
To the OP: If factory cases work and your reloads don't then its the reloads. If the reloads fit your case gage then they should be put together correctly.
So if they are put together correctly, and fit the gage, with the load you use, then your problem is your brass. Switch brass and try again.
I do question the "out of spec headspace" statement. What is out of spec? How did you measure the specs?
I was running sniper schools using M1C/Ds (30-06 for the NG, and M21s (308) for the RA. Some of these idiots would not pay attention in what ammo they grabbed. I found some weird spent cases and determined they were using 308s in the Garands. That would be out of spec headspace. But the only thing that happened was poor accuracy.
I don't buy the out of spec head space theory. All brass expands with its fired. its suppose to to seal the chamber. After firing it contracts allowing for easy extraction. It appears your brass is not contracting properly (hard to say without examining the brass, but a good guess).
Buy some new Winchester brass, reload it with the same load and bullet, and I bet you'll find you problem solved.
People over think headspace. Study Hatcher's Notebook on the subject and you'll find that a bit of excessive headspace isn't really that bad (except for accuracy)