when it dies, have you ever pulled a sparkplug wire and turned over the motor to be sure its still firing.
you need to this to insure its not sometype electrical problem that is shorting out your system after a short run time. new coil does not auto mean good coil, key switch, loose wire, etc: just be sure you are still getting fire to plugs.
probaby not: but at your present state you must suspect everything. most times when it does something as you describe, it's a stopped up carb. (normally when it will run better when you pull the choke (as you are doing with your hand over air inlet) you have a plugged up carb.
how long did you let the carb set in the cleaner.
did your carb kit come with new floats?
once I had a coil that would get hot after a short time and the engine would die, that is why i said to check electrial system first.
be sure to post what fixed the tractor as all of us may need the info in the future.
did you replace or clean the fuel filter in the carb. at elbow, it can be stopped up.
do you have a good flow of fuel to the carb, that you have a vented fuel tank cap