I,m not sure what would make it do this but if I drizzle a little gas in the carb it starts up then quits. ???? Does anyone have any ideas except a new carb.??
There could be a few issues going on here. I would guess the carb is not delivering gas, or at least not enough gas to continue the combustion process. Your "drizzling" gets things moving but not enough to keep things going.
The first Issue I would address would be fuel delivery to the carb bowl. If there is delivery to the bowl is there an issue with float or needle valve? Are the jets clogged? It does not take much to clog a mower jet if the mower has been left unserviced for a long time. Is the air filter clean? I sometimes start without an air filter on and spray some quick start in the carb to get things moving on a stubborn to start motor.
I don't think you would need a new carb without first trying to service the existing one. I'm not 100 percent familiar with your model Snapper, but I'm sure you can still get a carb kit. That's the first thing I'd try as long as fuel is getting to the carb and your air filter is clean enough. The age of your mower alone would make me suspect a bad float or needle issue. I'd pull that carb and clean it and replace the float and needle.
I'm sure you know this, but the cardinal rules are Air, Fuel, Compression, Spark. I'd put in a new plug if you have not already done so, check the compression, clean the air filter and/or replace it, and get that carb cleaned and serviced with a new float and needle, making sure the jets are squeaky clean.