I went out and ordered a new carb from Sears. (Sears doesn't have Briggs and Stratton list their engine type/number on their engines, so I couldn't oreder anywhere else).
After 10 minutes of popping the old one out and putting the new one in, it fired right up and did not stall. Ran smoothly and could restart after a shutdown without having to prime it. So success, and it only cost me $55. One interesting thing that the new carb has that the old one didn't is a adjustable vane in front of the air-intake. I left it wide open.
Thanks for all your help and suggestions!