I haven't heard of a distributor being temperature sensitive.
After the engine has warmed up and it stalls, take off the intake hose and get access to the throttle body. Confirm that it still won't restart. Then try spraying some starting fluid on the throttle body. Does it then restart for a moment?

What is the state of the electrical connectors at the distributor and ignition coil? Are the clips broken?
Do you ever get check engine codes? Make sure the light works (should turn on with key on, engine off).

Since it dies when you add an electrical load, maybe your electrical system is getting weak when the car warms up, but I couldn't say why.
It must not be your alternator since that wouldn't stall the car until the battery had also gone dead.

What is the condition of the wiring? Does it appear to be in bad condition at all? Has it been modified (aftermarket security systems, remote start)?
If there is a bad connection in the wiring, it will conduct less current when it gets hot.
If you can find electrical diagrams for your car, look to see if there are any fusible links shown that could be going bad. But normally, those just burn out cleanly like a fuse and don't recover when cooled off.


You could also look into the coolant temperature sensor. Some cars have a separate sender for the dashboard, so make sure you're looking at the one that talks to the ECM. You can test it by unplugging the connector and measuring the resistance between the sensor's pins with a multimeter. The resistance varies with temperature. There are charts which show the relationship between ohms of resistance and the indicated coolant temperature. I don't have the chart for your car, but it's probably out there somewhere. See if it's giving sensible values when cold and hot.