are you using the primer solenoid to start it or opening the carburetor butterfly some how to give it throttle when starting? If it only runs on the primer solenoid you can try this. Disclosure I am not familiar with older carburetors, but if its a 28 special from the 90s - one quick thing you could try is taking the drain plug off the carb fuel bowl with a flat head screw driver. Behind the drain plug there is an orificed jet that has a flat for a screw driver. You can remove that and see if its dirty at all as something quick to check and blast it out with carb cleaner. If you attempt this be extra careful pulling that jet out, they strip easy if you don't use a screw driver that fits tightly in the groove. ask me how I know....