Take a look at the end of your magazine tube, at the round threaded "nut."
See how there is no "finger" welded on the barrel?
To secure the take-down barrel of a Model 37, you twist that nut into a "finger" that is "welded" (it's actually machined )
Without that feature, your barrel must be screwed tight into the receiver, like a rifle barrel.
Here is a picture of my 1984 era Model 12
Don't Despair!
1. You can buy a used 12 gauge Ithaca Model 37 field shotgun on Gunbroker for under $300 after shipping, fees, taxes and FFL fees if you are patient. They are a screaming bargain. I would urge anyone to buy a whole shotgun rather than a replacement barrel. It's not 100% sure all barrels can be swapped, some will need 'minor' gunsmithing which is not 'minor' money and inconvenience.
1b. If you had a second shotgun, you could see if the barrel you purchased fit. If it did, you have two barrels. If not, you have two nice shotguns and can Ebay your extra barrel back for a small loss.
2. Your Deerslayer is worth a lot on the market, because I don't know why. But it is! The 'home defense' crowd loves them, because they are smooth as silk and tough as nails. As said, I would pattern the gun! I'd keep it, even though it's not my thing. If you must sell it, be sure to sell it on consignment through a reputable gun shop. If they buy it, they must give you less than 50% of what they can sell it for. Consignment... you keep 80%. With more posts in the Shotgun World community, you'll soon be able to post that gun for sale on the equipment board here. It's a real nice shotgun, I expect someone wants it.
3. For a field gun, I much prefer a barrel without the rib. The Model 37 Ultralight is different from the other fine Classic American Pumps because it's lighter to carry in the field. The rib takes away some of that distinctiveness.
Thanks for your post! I learned something new about Model 37s today!