Clean. clean. clean? Some of the newer 870's supposedly have rough chambers. This attracts dirt and allows the shell to expand into the small recesses. Try polishing the chamber with flitz or butchs bore shine.
I have seen the finish so thick in new 870 chambers I wondered how the gun got through QC like that. Wrap steel wool around a dowel. Splitting the dowel makes it easier to wrap steel wool around it. Then use a drill to remove the excess finish. If that fails you will have the barrel prepared for polishing as primer1 suggested.
A bunch of problems with crappy bulk 12 ga ammo---especially Winchester brand. Newer 870s tend to have rough chambers---a nice chamber polish job seems to cure this. My older 870s seem to have no issues with any 12 ga ammo. The cheaper "Express" models seem to have a lot more problems---especially the last few years production.
Move the patented and trademarked Remington Ammo Conservation Selector Switch to the "Off" position. It is located just alongside the "it's not the action, it's the pointability" decal found on 870 Express'.