I'd check to see if the problem may be that the hammer(s) are not retracting quick enough as the bbls are unlatched and tipped downward.
The cocking motion should instantly start to retract the hammer(s) off of the Ithaca's separate firing pins,,take the pressure off of them. (
This should be in any SxS,,separate f/pins or not).
There is no rebounding of the hammers so when they're down,,they are down hard on the pins and therefore the primers.
To see if the cocking linkage is worn and not taking up immediately,,dry fire the gun,,hold horizontally and open the top lever. Let the barrels down gently of their own weight till they engage the cocking linkage. You'll feel it stop after the initial opening.
I'd bet the breech is open more than a few .000 to perhaps a 1/32" or so.
That's amt the shell is dragged with the pin imbedded in the fired primer under mainspring pressure before the hammer is finally retracted to relieve that pressure.
The hammer(s) should begin to retract as soon as the bbls begin to lift from the breech.
The soft primer/hard primer thing does make a difference in that it will appear to cure the problem if the right ammo/primer is hit upon.
I just haven't quite figured out which makes it better and which makes the condition worse.
I've seen the charts and stats about which primer is which, but I've seen lot's of shotguns that have made the problem go away or at least get better,,and there doesn't necessarily seem to be any straight away choice of ammo. What works in one, will make it worse in another.
In any event, it doesn't take much to try a few different brands to find out.
Check firing pin length, shape also. Pins that are too long can be driven too deeply into primers. They can hang up then even if the cocking linkage is fine.
They don't need to be anything more than .045.
Squared points and sharp edges just add to any extra effort needed to extricate them from fired primers.
Just some thoughts to add to whats already been posted.
