Hey, welcome to the forum. Normally I am in the 2nd Gen forum, as I have a 97 RAM with the 5.9, but I have had your exact problems.
When it was in the process of setting that check engine light, did you notice any power loss? Like it would only rev to 3000 RPM maybe? Mine had about 185k on it when it did the same thing, #5 misfire, then it went to multiple cylinder misfires, etc.
Long story short, it could be a few different things, but maybe this will help speed up the search:
1. Clogged Catalytic converter(s). When mine clogged up, basically breaking the guts loose, then re-welding them together with the exhaust heat it clogged up the exhaust, making it impossible to flow any exhaust gasses through, and causing the misfire (resolution - replace cat)
2. Bent valve on #5. This is how mine started, and it took a complete teardown to fully diagnose it, but I had a bent exhaust valve on that cylinder, new set of heads cured it.
3. Might be some easy stuff like a plug wire, cap/rotor/plug thats easy to replace.
If you havent checked on these already, might send that info on to your repair guys.