Zuburg: Ideally, the owner of the street rod is a motor head and understands and appreciates your position as a guy looking for a good used engine. If he does, he won't be offended by your need to determine the condition of what he's offering to sell you.
Going for a ride in the '37 is a good starter. Pay particular attention to the oil pressure gauge readings when the engine starts up cold and what it reads at normal operating temperature. Note the oil pressure gauge readings at idle and when going down the road at different engine operating speeds up to 50 mph. Listen to the sound of the engine exhaust pipe(s) at idle; listen for an errant change in sound at idle. IF the engine seems to run okay from your test drive, I'd work with the owner to arrange to have a mechanic do further diagnostic testing on the engine for you.
In your circumstance, I'd turn to an auto repair shop you're familiar with or one recommended to you for engine diagnostic work. You need an experienced tech. that has the equipment and knowledge to do a vacuum, compression, and cylinder leak down test on the engine you're considering buying. This testing will tell the condition of the piston rings and valves. The oil pressure readings you gathered from your ride will depict the condition of the engine crankshaft and connecting rod bearings.
Since the owner of the '37 is an hour away from you, and since the car is unique, you may need to have the tech you employ travel to the owner's residence to do the testing on site.
All of what I'm suggesting you have done is to determine the condition of the engine which relates to what it's worth. Last November I had a knowledgeable car guy offer to sell me a used '56 Merc engine he's been sitting on for $1,000. His selling point to me was that it needed nothing (guaranteed!) and the rebuild I was contemplating on another engine was going to exceed 2K in my machining costs, replacement parts, and gaskets. IF I were in your circumstance, the offer of a complete, needs nothing, never rebuilt 312 engine, from someone I know personally, would appeal.
IF the previous owner converted your '56 'Bird from standard trans. to Fordomatic, it's likely that you've got a passenger car Fordomatic. When I did this years ago on a '57 'Bird, I fabricated a bracket to weld to the passenger car valve body control lever to permit connecting the 'Bird shift rod. The trans I used was out of a '57 292 equipped station wagon. The transmission kickdown lever required no modification to work properly with the correct 'Bird throttle linkage bellcrank.
There are some very knowledgeable participants on this Forum and numerous '55 - '57 'Bird owners. Others may chime in with their thoughts.
NoShortcuts
a.k.a. Charlie Brown
near Syracuse, New York