Note that when this gun was built .030"-.032" chokes would have likely been listed as Imp Mod. Full normally ran from 035"-040" or +. I can't recall ever having seen a catalog of this vintage which listed an Extra Full.
First I was ever aware of the term Extra Full was some Poly Chokes had that as a setting.
My 16 gauge Baker is marked "Baker Gun Co" on each lock plate & "Black Beauty Model" on the trigger plate. It does not say Special & does not have an F in the serial number. There is no indication to me that it was made by Folsom, nor would there seem to be any for your Damascus. I would come nearer suspecting the barrels to be Two-Blade Damascus than three at this grade level.