Oh yes, that’s one of those motors that is a reincarnated Bolens weed eater. If you sharpen the prop and hook a garden hose to it, it will do an OK, but clumsy, job on your lawn. Good that you found the ‘L’ and the ‘H’. But seriously, unscrew the ‘L’ low speed needle out of grandpa’s chain saw and see what shape it is, but I think yours is fine, just a little nasty. Since the plug is dry the mixture may be a little lean (causing the ‘cough’) so the engine is not getting enough fuel or maybe it is sucking air from somewhere. When it starts to conk out at idle, back out the ‘L’ needle and see what happens. Then, running again, screw the needle all the way in and see if the engine quits immediately verifying that your low speed circuit is at least functional. In addition, when it is conking out have you tried messing with the choke ?