You are describing a gas flow problem. Rust in tanks around the petcock can be deceptive with these vintage gems. Rust has a tendency to sflow to the gas line when the engine needs fuel, and then settle back when the engine dies Just as a simple test, disconnect the gas line at the carburetor end, then with air, blow in the tank to possibly clear rust particles away from the gas line and tank fitting, then reconnect everything and try starting the car. If this is the situation, and the car starts and runs longer, then you either need to install a small standpipe in the tank or do some kind of rust removal from the tank. I venture others can help with how to do that. This little test is EZ to do and will eliminate or identify the issue.