First, are you aware that you can get the factory service and parts manuals from the forum? They are in the TJ References forum. Here's a link to the thread containing them:
https://wranglertjforum.com/threads...ice-manuals-fsm-technical-documentation.4618/
Second, regarding sensors, there is only one engine coolant temperature sensor on your 2004. The PCM gets the signal and then sends it to the instrument cluster in a data stream.
Third, troubleshooting information on P1281 can be found on page 1839 of the pdf version of the 2004 service manual available on the page I linked above. Here's a screen shot:
That said, can you clarify your original post, please? You said, "this happened as soon as I cranked it yesterday" and "didn't allow enough time for warm up." Are you saying that you cranked it, it started, the CEL immediately illuminated, and you immediately checked the codes and found P1281? Or did it run a while before you checked codes? I ask because, according to the screenshot above, that code is triggered when you drive for at least 20 minutes with the engine coolant temperature failing to rise above 176°F. If it triggered immediately, the PCM may be bad because the PCM sets that code and shouldn't set it immediately. If it wasn't immediate, you should have noticed that the temperature gauge didn't rise as the TJ ran. In that case, it's either a thermostat stuck open or a bad sensor. The thermostats are much more prone to failure than the sensors because they are mechanical devices in a wet, hot environment.
Finally, if you put your TJ information in your profile, you won't have to tell the forum what you have every time you post a question. Most of us know to look at your profile for more information. I did so when you first posted, didn't see any info, and then got distracted by other forum activity. I would have answered yesterday if the info was in your profile.
Good luck tracking down your problem, Brian!
