I like the suggestiion of electric problem, crank sensor failing when hot or ignition transitor, but I'm more on an induction problem. That is the air getting into the engine to burn the mixture
Because you say it runs better in neutral, without load on engine,(transmission in gear), any air duct leaks or vacuum lines cracked on high mileage cars, can cause the engine to run rough. Once you start accelerating, the air volume goes way up (throttle open ) and leaks are not as noticable and compression goes up.
Compression is always a good place to start checking on an old engine. If the air volume or compression pressures are not correct, it can cause an engine to run poorly at idle. The engine is a big air pump and it needs to be sucking air without any leaks to run right . Any leaks at idle, or an engine with low comprssion will throw the settings off. A worn timing belt will drop off your compression readings.
Squeeze the duct going to the engine from the air filter, if it shuts off the duct its cracked somewhere.
You say it runs better with part throttle?