I am going to say it is a carburetor issue. It is possible that you have gotten debris in a passage in the carb, partially blocking a circuit.
By your description, I would guess that you are running on the lean side at part throttle, hence the stumble upon hesitation. Make sure your accelerator "squirt" is good on both sides. Remove your idle mixture screws and blow them out with carb cleaner. Reinstall them to whatever the spec might be, then start the engine. Adjust each idle mixture screw in until it changes the engine RPM slightly, and then back them out another 1/2 to 3/4 turn. There may be an actual Studebaker spec for this, but if not dont worry about it.
Drive it and see if the hesitation is gone. If you still have a bit of hesitation, turn each idle mixture screw out another 1/4 turn and try it again.
Keep in mind that a lean running engine will run hot, so you may kill 2 birds with one stone.
Good luck, hope this helps.
Posted on Jun 11, 2012, 9:56 PM from IP address 126.96.36.199