I recently bought a 40 year old Browning BLR in 308 and I had all sorts of problems getting cases to chamber both smoothly and completely.
If I relate the issues I had you can decide if they mirror any of the problems you have.
All the brass I reloaded was sticky, so much so I bought 20 rounds of match ammo which cycled perfectly, can you get factory loaded ammo to try?
The brass I was using had been fired in my previous 308, a much loved Ruger #1, as such I'd only ever neck sized it.
I'd been using max loads of CFE223 behind barnes mono bullets.
Even though I'd full length resized prior to loading for the BLR, the stickiness was there. A quick pass with a micrometer showed the brass was springing back, the issue was not as severe with federal brass but was with ex military ie palma match brass.
I found that at least 4 passes through my LEE FL die were needed with 7.62 brass as opposed to 2 passes on 308 brass before the base diameter got within 0.001" of the saami specification.
I realise that my issue was largely brought on by myself so I pulled the decapping/expander mandrel out of the die and resized the 40 odd reloaded cases I had, same result, 4 passes for 7.62 brass, 2 for federal, PPU and Winchester. All function OK now about a week after I performed that job so the cases haven't "creeped" back to an oversize state.
I'd say try multiple resizing operations with plenty of case lub round the base to see if you have that issue, it's so easy to do on your 5 cases before you start polishing the chamber etc.
nato brass on the right, federal on left, federal has been reloaded in excess of 10 times, nato will be 3rd reload. Apologies, unable to show before and after sizing, just after.
Just my 0.02$ worth.