It could go by the make and serial number of the mower but you could
take the carb with you also.

They wanted the stamped number off the engine block. I had the model
of the mower but no help as it's old.

Sounds like the engine is dying because it's starving for fuel (unless it makes black smoke) when the governor pulls the throttle up and that could be a sign of low fuel level in the bowl (float/needle valve problem) You probably should call the parts department and ask the exactly what they need to look up a parts list for that carb.

I would of checked the float out better but didn't have the
replacement gasket and the float is attached to the top part of the
carb housing and some of it extends below the gasket so I couldn't
remove it without scraping off the gasket. The local shop (and I
trust these guys) is open tomorrow 8-1 so I am running by there. Good
idea to bring the carb. Two bolts and it's in my hand. My Dad (who I
got the mower from) just told me that he thinks it's carbon buildup in
the heads and/or the exhaust system. I kind of doubt that as that
would cause it to run bad all the time and not just when going under
load. My hunch is pointing me towards the float as even when it
starts to die if I spray starter fluid (or dribble some gas) directly
in the carb it stays running until the fuel is used up. The float is
not in a seperate "bowl" with a releif spring screw plug in the bottom
like many small engines, it's in a machined out section of the bottom
housing and only has one hex plug in the side and no drain in the
bottom.

Thanks for the advice, I need it :O)
Scott