Hello, Patricia;
Intermittent failures can be very difficult for a mechanic to nail down sometimes, particularly if there are no obvious clues to help in the diagnosis. The onboard diagnostics used in vehicles of this age are not nearly as detailed or informative as they are on newer models. For example, a 1996 or up OBD-2 vehicle stores hundreds of service codes and when this happens it also stores a snapshot of engine data to aid in diagnosis of intermittent failures such as you are experiencing; a 1993 model such as yours only stores a few generic codes and no data. This is why diagnosis can be so difficult sometimes.
There are literally dozens of possible causes for the symptoms you are experiencing. It is also possible that there may be more than one fault in the vehicle.
The fact that it starts up after cooling down for a couple hours is interesting; this is suggestive of an electrical component failure. Often, electronic components can experience thermal related problems. Likely possibilities are the ignition module, ignition coil, engine control module, and crankshaft position sensor. If your car has a mass airflow sensor, that too is a possible cause.
Whatever component is causing the problem will likely get worse; the best thing that could happen would be that the engine would shut off and stay off; that would make diagnoosis easy. It does make the vehicle unreliable in the meantime though.
Alternatively, you can try to induce the fault to try and identify the failing component. It seems as if it may be heat related; try taking a heat gun or a hair dryer and heating electrical components to see if you can cause the engine to stall. In particular, heat the ignition module and the engine control module (computer). General motors ECM;s were well known back around this time for developing poor solder joints on their circuit boards that would result in all kinds of erratic random intermittent symptoms, such as stalling problems, random service codes, and check engine light problems. I believe the ECM is probably located inside the vehicle up behind the glove box on this model; it is an aluminum box about 6 x 9 x 2 inches with either 2 or 3 large electrical plugs going to it. Unbolt it from it's bracket and try shaking it, gently twisting and tapping on it, and heating it with a heat gun to see if you can make the engine do anything unusual; if so, it is likely faulty and possibly the cause of your problem. If you replace it, you will have to swap the internal calibration assembly (memcal) from the old one to the replacement; it is located under a removeable cover and contains the software calibration for your particular model.
If your model has a mass airflow sensor located in the air inlet duct leading to teh engine, try tapping on it with a screwdriver handle or something similiar, to see if the engine stalls.
You can also try shaking the wiring harnesses around under teh hood to check for poor electrical connections or broken wires inside the harness. If you are not familiar with repair procedures, it would still be extremely helpful to any mechanic you take the vehicle to if you can determine that moving some particular wire or heating some particular part makes the engine stall.
If nothing else helps, you may want to have your mechanic try installing a new crankshaft position sensor to see if it helps; they are pretty inexpensive and can cause intermittent stalling problems. Next, if trying the shotgun approach, I would suggest possibly a used ignition module and coil assembly, and perhaps a used engine control module.
There is no quick easy magic bullet for finding this type of intermittent fault; often some luck is involved.
The code 44 you have indicates a lean condition. Checking fuel pressure might be informative; you may have a failing fuel pump. Or, it could be a result of an ignition misfire or control problem as described above. I think I'd try to locate the stalling problem first, before chasing a lean running problem....
I hope this is helpful to you; if so, an accept would be most aprpeciated! thanks!
