....sounds like the vendor just tumbled them to clean them. When a website claims "Processed", need to verify what their definition of "processed" is, if they detail it. The claim of "Processed" can vary greatly, but at a minimum it will be cleaned, either chemical wash or tumbled, primers still in the cases. Fully processed will be decapped, sized to SAAMI spec, trimmed to SAAMI spec and then cleaned (dry, wet, washed). Regardless, any 1x fired surplus brass should always be sized in YOUR dies that are set to YOUR chambers dimensions and trimmed to YOUR chambers dimensions before use.
Annealing beforehand will certainly ease the shoulder bump process and normalize the neck tension, a single pass is all that's required. That 1st time sizing will seem "harder", but after that and fired in YOUR chamber, the amount of force to resize will ease up.
A small based sizing die isn't always needed, but you can check that by using one of the resized case to make a dummy round and see if it will chamber properly.
Caveat: An insufficient shoulder bump can prevent full chambering and give a false reading that makes it appear that a small base die is needed. An overly long seated bullet can jam in the lands and do same. Individual bullet style of ogive is a factor when choosing a seating depth,
...FWIW, an invaluable tool I've been using for multiple calibers is the Sheridan Slotted gauges. Takes all of the guessing out of trim length, shoulder bump and bullet seating, I can't recommend it enough --->>
https://sheridanengineering.com/product/308-win-ammunition-gauge/