I used to be a dealer for those. Great trimmers especially. Almost every one we sold never came in for repairs. So much better than whats out there today.
Alot of the old murray diagrams never made it online. I still have my old micro films with break downs but idk how to get them on here for you. Maybe try to reach out to a local shop in your area.
Have you tried resetting the position of the vari pulley? That usually needs to be done if you replace the belts, or the machine stalls while moving, runs out of gas. Put the machine on a flat level spot so it wont roll away. Then get on it start it up. Make sure your shifter is in neutral. Then...
Id price a carb assembly. Ive gotten some honda original carbs for alot of the gcv engines for under 30 dollars. I only try rebuilding them if the new oem is crazy expensive.
I would pull the valve covers for sure. Those briggs twins are know for having the rods bend and sometimes they fall into the engine block. For some reason alot of them will run smooth on one cylinder until they get under a load.
Almost all single cylinders have issues do to them mostly being made in china. I see less issues with both briggs and kohler twins but i usually recommend a kawasaki.