Given what little I know on the 350's, my first guess would be an iffy spark plug. That's cheap and certainly won't hurt to replace it. Get a new NGK plug and see if that fixes it.
If not, next place I'd look would be partially restricted fuel petcock. If it's slowly running into the carb, and you're running it a hair faster than fuel is coming in, eventually you run out. After a few mins enough fuel trickles back in and it will run again for a bit. Also take your gas cap off. If the vent line is plugged it will create a vacuum and not let fuel run to the carb.
Lastly, could be electrical like described in this thread.
Look in your right rear wheel well, and you'll see the regulator. They can partially burn out and do what you describe. Next time it does it, try tapping on it with a screwdriver handle and see if it will start.
My guess is a plugged up fuel tank cap vent that traps a vacuum - stops fuel outflow from the tank. This is very easy to check.
Or, your valves are too tight. With not enough clearance when cold, it becomes no clearance when hot. The valve gets ever so slightly longer and thus stays open when hot. An open valve means loss of compression when hot. It runs poorly or not at all until cool again. You can also burn a valve seat with this problem.
I too think it could be a bad gas cap vent, VAPOR LOC, will do this as things warm up
so will ALL fuel flow issue's, HOW old is the fuel itself too??
simple problem shooting steps here
start with drain old fuel, inspect peacock, clean if needed, (also maybe screen in tank is plugged up??)
pull or better yet, replace fuel lines from tank to carb, there cheap and easy to do and rule out one maybe being clogged up?
once this is done, replace fuel with GOOD fresh fuel and see
if still there check cap for venting, when it dies, OPEN the cap and listen for air rushing out/in!
if that doesn't fix it, replace plug
if that doesn't solve it
come back and let us know and more idea's will sure to come! LOL