So...I have actually been cruising some other forums (on other sites) for this issue and found a couple of things. Here is an excerpt of text that seemed to resonate with me and something that I will probably try. The description this guy gave of his 930 FTE issues sound very similar to yours and mine:
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"I finally decided to put the gun under the microscope to see where the problem lied.
I found that the chamber was rough as could be. I ran my finger around in it and I could feel the scratches and grooves in it. That's always a problem.
Also, I noticed that the extractor wasn't really grabbing the shells worth a crap. It seemed to not reach in toward the bolt face enough. I fixed that too. I had to remove the extractor from the bolt and file down a little area where the extractor met the bolt body. this allowed the extractor to reach in towards the center of the bolt face a little more, ensuring that it has a good bite into the shell casing. This also gives a slight bit more pressure on the shell pushing it to the left hand side of the gun during cycling ensuring good engagement on the ejector, that makes sure that it throws the old out with authority and gets it out of the way for the new cartridge instead of jamming like mine always did. The extractors seem to be like rough cast, or rough forged and that allows for alot of variance between production runs. Some may need help. Re hardened the steel and reassembled.
Lastly I noticed that there are two sets of gas rings in the gas piston. Each one has its own gap. Ensure that the gaps are opposite of each other, just like piston rings in an engine. This will reduce lost compression, or in the case of the shotgun lost gas pressure.
So, polish chamber, align gas rings and adjust the extractors attitude and you should be on your way to a happy gun.
I did all this in about an hour, my gun instantly ran like it should after the fixes. Im 100% positive that the three fixes I did directly improved the operation of the gun."
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I don't have another 3-gun comp until the middle of February, so I think I am going to get out the dremel, polish the chamber, check my gas rings & extractor, giver another good cleaning and go out with some assorted ammo (inc. the low-brass cheapys that I have been having problems with) and see if I can get this thing cycling again.
ONE LAST THOUGHT ... since this guy is talking extractor, I wonder if just a really good cleaning of your extractor would help. I have an old Remmy 550, 22LR and all kinds of gunk gets built up under the extractors and makes them very ineffective. They aren't easily removed, so I just have to soak my entire bolt in gun cleaner and then it is good to go for another 1000+ rounds ... until it starts FTE'ing again. Maybe your extractor has a lot of build-up under it?