1947 is correct. here is a link to serial nbrs by dates on the Ithacagun website
what you have is a nice old, late 40's Ithaca 37 12ga. Well worth hanging on to.
did you measure that 29 1/2 from the front of the reciever ? if so, then that is a 30" barrel (measured overall).
The barrels on Ithacas became interchangeable after serial nbr 855,000.
Prior to that, they were individually fitted to each reciever. If you look at the front flat of the reciever, next to the barrel, you will see the matching serial nbr. The barrel is serial nbr'ed to the reciever, often times/mostly with a -2 -4 etc indicating the choke, along with Full, Mod ImpC etc.
The magazine nut turns to the right to loosen. Do not use pliers on it. The style of barrel nut you have is hard to come by. I do know where there are some pristine ones if someone took a pliers to yours in the past.
DO NOT try to put a new(er) barrel into your reciever, it will ruin it. Pre 855,000 the "interupted threads" that allow the barrel to come off, actually had a pitch to them. Post 855,000 are straight cut, with dimensional interchangeability from gun to gun.
The reciever is unique to the 12ga size, the 16's and 20's are built on their own frame.
used barrels can be found, but they have to be fitted by a gunsmith familiar with the design,
is operated by the former head gunsmith of Ithaca before they were bought out and moved to Ohio. He specializes in the old guns.
Don't cut your original barrel back, take the time to find a takeoff and have it fitted. That way you will preserve the collector value of the gun. Same with the buttstock. There are some old buttstocks on eBay and gunbroker if you want to have a recoil pad put on. I also know where to get semi-inletted new ones.
what kind of questions do you have ?