I believe your problem is a carb which is not getting enough fuel. If I had it I would remove the carb, give it a good bath and install a new float needle and seat. Briggs part number 398188. Here is a website which will show your engine parts and a description on how to replace the needle and seat.
Illustrated Parts List
Needle and seat replacement
Remove the carb, and then remove the float bowl. Check the float bowl jet (which is the bowl screw) and make sure the jets both horizontal and vertical are clean and open. Tip the carb upside down and remove the float pin and float with needle attached. Look in the float needle passage and you should see the red float seat at the bottom of the passage. This is where a #5 crotchet hook would come in handy as you need to remove this seat. If you have no hook, but compressed air, you can blow through the fuel inlet and try to pop the seat out. Put your thumb over the passage to prevent the seat from flying who knows where. No air or hook try bending a stiff paper clip to dig the seat out.
With the seat out clean the passage way with carb cleaner. Now you must find a drill bit slightly smaller than the passage way, to be used to press in the new seat. Apply a very, very small amount of a very light lube to the new seat. 3-1 oil or lighter, to help seat it better. Carefully insert the new seat in the passage way with the rings on the seat down toward the carb body. Slowly and carefully force the seat down with the back end of the drill bit. Once it is seated, check to see that it did not flip and the rings are up. If everything looks correct, attach the new needle to the float and install with the float pin centered. It everything is correct, the float should seat level to the carb body, when looking at it upside down. If everything looks good reattach the float bowl, making sure that both the bowl gasket and the nut gasket seal properly. Reinstall on the engine and test unit. Remember to have patience and take your time. Good luck, but I don't think you'll need it.
