First make sure it is not the low oil sensor shutting it down. The five or so second delay after starting before shutting off is just what the oil sensor will do if oil is low or if sensor had failed. On opposite side of engine from Carb is a yellow wire coming out of the block down low. Trace that yellow wire a short distance up to the connector, often covered with a rubber sheild that slides out of the way, and unplug that yellow wire. That will disable the low oil sensor. Then see if it still dies.
If you still end up with a need for a new carb, the first place to check is right here at PERR's home page, Bruce can fix you up.
Nevada! The last free state!
Retired Briggs MST, Honda Top Tech.
Currently full time experienced novice fisherman.
You should've been here yesterday, the fish were really biting..