Thanks for all the help everyone.
I got lazy and ordered a carb for $14. Problem solved, it no longer shuts off.
But I have a new problem. It has no balls whatsoever. The engine bogs down as soon as I apply load (thick grass). I have to mow super slow for it to stay running, though it does...
I'm completely stumped as to what is wrong with this thing. It's a SEARS craftsman eager-1 4.5hp 22" engine model 143.964502.
It starts on the first try, runs for as long as 10 minutes, and as soon as I'm cutting a thick section of grass it starts surging and then dies. Then it won't restart...