Sounds like the high speed and curb idle settings need adjustment, and
possibly the fast idle cam linkage that connects to the choke.

The first thing I would do when the engine dies out is look to see if
the idle speed screw on the throttle cable side of the carb is touching
the throttle lever. If it isn't either the screw is backed out way too
far, or the fast idle speed screw is still controlling the idle speed.

My guess is that what's happing is that the fast idle linkage is out of
adjustment, and eventually the fast idle screw loses contact with the
fast idle cam, and that is when your engine dies out, even though it
sounds like it is staying on the fast idle cam longer than it should.

Turn the curb idle screw in 1/2 turn or so and back the fast idle screw
out several turns for now, just to get the curb idle screw in contact
with the throttle linkage.  Start the engine and warm it up, then set
the idle speed and mixture properly. your carb kit instructions or your
owners manual should have the specs for setting the idle and fast idle
speeds, but you may have to tweak the fast idle speed somewhat. Every
engine likes it a little different, and climate is a factor also.

After the curb idle speed is straightened out, then set the fast idle
speed. This is normally done with the engine warm, and the choke open,
but you will have to manually move the fast idle cam linkage to get the
fast idle screw on to the cam. the manual or instructions should tell
you which step to set the fast idle speed on. Before you do the fast
idle speed though, I would double check the linkage adjustment per the
carb rebuild kit instructions. It sounds like there is a problem there.

Chris