It isn't off topic at all. There has been a version of that gun in my safe for the equivalent of five dog lifetimes. The right one is indeed a fine grouse gun.
I'm not sure when the Ithaca company went to the cheaper version of the buttstock for the 37 maybe 1956 or so, but, they are truly ugly compared to the earlier efforts. More than one 37 has had it's pistol grip rasped off, but, last I checked the folks at Upper Sandusky will sell you a brand new copy of whatever version you want, so you don't need to sacrifice the original. Ditto on the safety. Hang onto the lefty version, someone down the road might need it should you decide the 37 isn't your cup of tea.
Remember to pump! Skeet goes a long way toward keeping you sharp on pumping your gun. Yes, I have forgotten to pump a gun in the heat of the moment, likely from my mostly complete conversion to double guns, over the years. If you will be hunting woodcock, do remember to plug your gun. I have used a pencil, in a pinch. The barrels on the guns of your era are NOT interchangable. You will probably be working with the barrel you have, either relief of the existing choke, or Mike Orlen can set you up with reasonable cost choke tubes.
A long time ago, I used a Remington 17 (same gun, different name on it) to take a limit each of grouse and woodcock on a sunny afternoon near Orr, MN, and became camp hero for a day. I never shoot more than a brace of any birds these days, but, a 37 would be more than enough gun for the task.
I think you will enjoy it.
There are many fellows here who enjoy pumps along with their doubles. You should, too.


Best,
Ted