if teh battery is more than 5 years old then I'd suspect the battery. If the battery is newer than it's likely that the cables are the problem.
If the cables are dirty of have some crusty build up on them, take the cables off the battery, clean the battery terminals and clean the wire terminals. If there is any white or yellow/greenish crust on the connections they are definately dirty and need to be thoroughly cleaned. In severe cases the cables may need to be repalced.
As for your engine dying problem it is fuel related. I don't want to say that it is a problem with the fuel metering in the carb becasue once you choke it its good for a little while. If it was a fuel metering problem it would always be there as long as the choke is off. I'll go with poor fuel flow to the carb.
You're on the right track. Re-building the carburator and making sure to clean all the parts before you put it back together will make sure that the carb is working correctly. But, I also suggest changing the fuel filter at the same time. If the fuel filter and the carb rebuild don't fix the stalling problem then its probably the fuel pump.
The higher horsepower Briggs engines like your 18hp one, have diaphram fuel pumps that work by the air pulses generated inside the crankcase as the pistons move back and forth. The pump is probably mounted to the engine, near the top and is probably square in shape and has 2 fuel lines going to it and one hose that goes to the crankcase.
These pumps are notorious for going bad on engines older than 10 years old because the modern fuels have additives that weren't present back in the day and they tend to harden the diaphrams which loweres the ammount of fuel they can pump. Replacement diaphrams are made of a material that will resist hardening. Usually the pumps are pretty cheap and for a few more dollars than the rebuild kit you can get a new one.
