Hello and welcome to JustAnswer!
Your problem sounds like a weak heater on the upstream oxygen sensor. This is a problem common to all Jeeps, trucks and Ram vans from '92 to 2003, anything that used a JTEC engine controller.
The oxygen sensors have a built in heater because the exhaust doesn't heat the sensor enough to properly monitor oxygen content, and the heater also helps it get into closed loop operation quicker from a cold start.
The sensor has a 5v pullup voltage on it for circuit diagnostics. When the key is first turned on the sensor reading will be at 5v. As the heater starts to work it bleeds off this voltage until the voltage is in the normal operating range.
When you start your engine it's in open loop mode. This means that the oxygen sensors are ignored and fuel control is managed basically by software alone. After it warms for a few minutes it will go into closed loop, when the oxygen sensor readings are then used. If you have a weak heater on your upstream oxygen sensor the pullup voltage will be bleeding off too slowly, and when the truck hits closed loop the sensor is still reading the higher pullup voltage instead of the actual sensor reading. The engine controller sees this and thinks its running rich and leans it out, giving you rough running, hesitation, and often some lean backfiring in the intake.
Also if it's already warm and you park it and let it idle for a few minutes the sensor will cool back off. The longer it sits and idles it will run worse and worse, usually you can hear the idle air control motor opening farther and farther to keep the engine running, giving a large sucking sound under the hood. This is more likely to happen in cold weather.
You would think this problem would set a code, but the software used in these engine controllers let this slip by without setting a code normally, because the voltage is still within normal operating range (not shorted to ground or an open circuit). It simply believes the wrong voltage is the true signal because it's still within it's voltage window of not being shorted or open.
I'd recommend replacing the upstream oxygen sensor, the one on the front of the catalytic converter. After you do, disconnect the battery to clear the stored fuel control adaptives out that have been falsely learned by the bad sensor. This step is just as important as replacing the sensor, since the controller has been adapting fuel lean because of the incorrect signal.