Hello and welcome to JustAnswer!
You're correct - your truck doesn't have an EGR valve. They went away on these engines after 1997.
Your problem sounds like a weak heater on the upstream oxygen sensor. The sensor has a built in heater because the exhaust doesn't heat the sensor enough to properly monitor oxygen content, and the heater 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 may 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 worse in colder weather and happens more in the advanced stages of heater failure, you may not notice this symptom.
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).
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.
Sounds like we have bigger issues than it just backfiring through warm up.
Does it seem rich, lean? Has the engine light come on?
These new symptoms seem to have started as a result of replacing the oxygen sensor, correct?
Where did the new sensor come from/what brand was it?
Let's disconnect that new oxygen sensor, disconnect the battery again to clear the adaptives and see how the truck runs. If it runs ok then return that Bosch sensor and either get a Nippondenso or a Mopar sensor from your dealer. Bosch sensors are a pretty big source of problems.
It doesn't take long at all, thirty seconds or a minute will be plenty.
Too bad about the wrong sensor. You have to watch them parts guys! Let me know what happens Monday when you get the right one in there!
Originally you said the problem was only for the first 2-3 miles after a cold start. Is this still the case or is it more than that?
Did you replace the upstream sensor with the Mopar now?
Has the engine light come on at all yet?
Did you disconnect the battery this last time after replacing the correct sensor?
Do you have four oxygen sensors on your truck or two?
The first thing that needed to be done was determine if you had one or two upstream sensors. If you had two upstream sensors then either one of them could be causing the problem and they would have needed to both be replaced or monitored with a scan tool to see which one is bad.
Slide under the truck and see if there is a catalytic converter on each side of the engine. If there is a converter on both banks then each one will have an upstream and downstream sensor. Whichever bank you were just on, go to the other side of the truck and see if there is an upstream sensor there.
Let's unplug it first and see what happens. Unplug the sensor and make sure the wiring is out of the way so it won't get burnt on the manifold. Do yet another battery disconnect, take it for a ride and see how it does.
Nothing else has to be removed, but the upstream sensors can definitely be a pain to get to. I can't recall exactly where the connector is on this year/engine combination. Is the connector above the trans?
That's great to hear!