I didn't know they made one of these in 20ga, but if a trap model exists, I bet it would be a neat curiosity.
Many nice days spent in the field with a 20ga 37 when I was in school. That nice day got a little hiccup when I pulled the trigger, because at 5.5 lbs, it kicked the fillings out of my teeth. But a neat gun anyway. That little orange Raybar sight was the look of death.
There are many factors that determine Model 37T values. Pre-war or not is probably the biggest. Was the game scene hand chased is another big determination. Being all original and condition like any gun matter as well. Buying a mint condition field grade M37 for $400 isn't a bad price. The target grades are worth much more. There is a decent market for the higher end Ithaca M37 and a lot of us collectors are out there. I recommend Walt Synders book "Ithaca Featherlight Repeaters...The Best Gun Going" for anyone looking to learn a bunch about them.