I once owned a factory original Sterlingworth that had shallow rim cuts in its extractor. The gun acted just like yours, tight closing with shells in the chambers and forcing the barrels off face. I had a piloted countersink made up at a tooling shop and deepened the rim cuts by hand. PS, I don't have it any more, loaned it out and never got it back, long story.
I don't know if this vignette will help you because you say you have new barrels. I didn't catch if they were extractor or ejector. If you're using the extractor or ejectors from a different set of barrels their rim cuts may not align with the cuts already in the barrels, same scenario as with the SW I wrote about here. If the new barrels came with the extractor or ejectors I would suspect the rim cut depth before I would a tight chamber. Just my own thought process.... hope this helps.
If you are handy with tools you might check Brownells shotgun chambering stuff to see if they have rim cutters and/or chambering reamers with integral rim cut. I haven't dealt with Clymer so I don't know what they have. Sorry, I don't have the name of a good gunsmith in the NW. Silvers
