Have a pull start mower that was working fine until today. Got it
started no problem (about 3 pulls) and got about half my lawn done.
Stopped it twice to empty the bag and after the 3rd start (both 2nd
times started right up) it was working fine until I hit some heavy
grass and it bogged down I pulled back but it stopped. But when I went
to restart it it started fine but with a few seconds it started to slow
down and then stopped. Working the throttle doesn't seem to help (if I
feather it it will keep running a little longer, but still stops). One
thing that is interesting is if I rock the mower back and forth or
side to side it will rev back to normal speed so something tells me I
have some sort of fuel flow problem. Let it sit for over an hour with
no change.

Cleaned air filter, oil looks good, had just filled tank with fresh gas
(although when I started it it seemed like some gas did leak out from
under the cap).

Unit is 12 years old.

Any ideas on what to check?

Thanks