2003 Lincoln Town Car keeps running after engine turns off then cannot start again

Are you sure it's not the fan? Pop the hood next time it happens. The fan is not controlled by a relay but by a fan control module, which in turn is controlled by the powertrain control module. Power to the fan is through maxi-fuse number 12. There aren't too many motors on your car that could drain a battery this quickly, so the fan motor seems most likely to be the culprit - although what is causing the fan to run (fan control module, or PCM) will have to be determined.

Thanks so much for replying. This is my parents' car and I've never been present when it has happened. It's been to two separate garages (dealership twice and local garage once). Unfortunately, after an occurrence they wait overnight and jumpstart it and bring it to the mechanics where it never happens for the mechanic who finds nothing despite checking the fan(s). Frustrating because the car is in great shape but now leaving them stranded. They would prefer to fix this rather than give up on the car, but they need it to be reliable especially headed into the winter.

Let's assume the fans are fine mechanically. The problem then is likely to be what's controlling the fans, causing them to continue running long after the car is turned off. They are designed to run for a few minutes after shutting off an engine, but then shut off. I don't know if a diagnostic test can be run on a fan control module or not, but they can do that for a powertrain control module. Or perhaps the PCM can be reflashed back to its factory settings. The problem may be a temperature sensor or an intermittent short/open with a wire harness as well. Unfortunately your shops have taken a "let's wait until we have a hard fault" approach to your intermittent problem. (Keep in mind I'm assuming the problem is with the fans running, based only on your description of the fault.)

@KenF thanks for the response my mother is going through the same problem as I type and I needed answers