A 20ga chamber should taper from about .700" just ahead of the rim down to about .685" just where the forcing cone begins. I don't know how you are measuring but somehow doubt your caliper readings. If the chambers are indeed .670" they weren't finished, & I'm surprised you can even get a shell in. The pins look too pointed & perhaps too long as well, primer indents are certainly deep enough. I also suspect the timing of your cocking levers is off & hammers are not beginning to cock on initial opening of the gun.
I agree with 2-P on all points. Funny how often that happens. Maybe because he's usually right in his answers? The primers are protruding because the rim cut is too deep. The rough chambers add to this problem by allowing the shell walls to grip the chamber walls so the pressure pushes the primers back. The primers then wrap around the too long and too pointed firing pins. The primers also protrude into the FP bushing spanner holes, causing more drag. Like 2-P suggested, the first things I would do is shorten the firing pins to protrude about .045" at most and round them nicely. Then do what is necessary to make the cocking levers or rods start to cock the hammers slightly sooner. The too-deep rim cuts could be made shallower by cutting them deeper and installing rings of steel permanently. This might cost more than the gun is worth. Maybe Mike Orlen does this. If so, he will be less expensive then most. If you just bought the gun chances are the seller had new pins installed and they were not properly rounded and fit for length.
Two more points: the problem is not that the primers are not being seated deep enough. The primers are backing out when the shells are fired. Proof of this is that the battery cup is flush with the rest of the head.
Other poing: remove the firing pins to round and shorten them. Trying to "stone them in place" will make a mess out of the standing breech face and never do a good job. You may have to make a spanner to remove the fp bushings but do this the right way, with the pins out of the gun.
