Kyle, check the points push rod and see if it's the fiberglass type. If so, change it out for a new one. I'm willing to bet the old fiberglass rod is worn and frayed, soaking up oil which will cause it to swell, and isn't sliding in and out like it should be, which will cause your engine to run fine for a couple of minutes, start sputtering, then eventually shut off. Let us know your results or findings.
Just so you know, I have used a 16 penny common nail, cut to the length of the original push rod length, filed the ends so they were slightly rounded, and haven't had any issues at all. Just throwing that out there for you, as I don't think push rods are available anymore through Deere or Tecumseh.