I've owned a number of John Deere two cylinder tractors, both gas and diesel. I've rescued long abandoned tractors, rebuilt them, worked at Deere dealerships, and actually did quite a bit of farming with them. Not once did I encounter an issue using detergent oil, I had a number of friends and neighbors who had older tractors and they never had an issue switching to modern oils in tractors that got used in real farming, I believe you will experience no issues using modern oils.
The Wix/NAPA filters used to come with two rugger seals for the oil filter cap. The smaller one fits your tractor, the larger one was for the later numbered tractors such as the 4020. In my experience these seals did not require replacement all that often, but generally did if the tractor hadn't been used for some time. The old seals get very hard, I've had to use a very small chisel to get the old seal out
If you are worried about sludge, the crankcase cover is right there and easy to take off. Take a look in there and marvel at the size of the rods. I've found a few odd things down in there, ranging from rocks to coins. I've never seen any sludge in the crankcase. There could possibly be a bit in the valve cover, but I've never seen enough to worry about.
I tended to use SAE 30 diesel oil for everything, in the winter I'd use 10W. Later I tried 15W-40 but not for long. It used to cost a few cents per gallon more than straight weight, and I experienced no benefit.
The 1937 may still have babbitt rod bearings, taking the cover off would give you the opportunity to take a look at them. There is a grease point at the front top of the fan shaft, these seem to be frequently forgotten.
Good luck!
