Sounds like maybe you had the a/c on and the compressor was trying to kick on but the high pressure switch was kicking it back off due to high head pressure? Anyhow, make sure your a/c is off while doing this.
The '98s came out with a soft seat thermostat because people complained about the movement of the water temp gauge as the good thermostat opened and closed. The soft seat thermostats proved to be a problem and a revised thermostat was issued. Best to get a good thermostat from Cummins if you're going to change it.
Sounds like your fan clutch isn't working. My Cummins powered Rams would kick on the 22" fan with a loud roar in hot weather towing and around town in heavy traffic. When you did the cardboard test and the temp came up on the gauge, did it also come up on both right and left sides of the radiator? If the water pump isn't up to snuff or something is restricting flow, the radiator won't get hot which in turn engages the fan clutch. It sounds like your clutch is bad from what I've read. I bought a fan clutch wrench kit from PepBoys for my '98. One of the Ford kits works but at the time their wasn't a kit specifically for the Dodge. I also bought a severe duty clutch for about $70 if I remember right. Mine was working but I was chasing a vibration that turned out to be the hydraulic power brake booster hoses putting normal 2100rpm engine vibes in my steering wheel and peddles. My '95 didn't have the hydraulic booster so the vibe was new to me. You don't need to take the fan all the way out to change the clutch. If you do, you'll have to walk the fan out toward the passenger side. It's very tight and will take some patience.
I'm going to see if I can attach a couple of pics from my picture host site.

 

I had some overheating problems for the first time this summer pulling my TT up some very steep passes in OD with TC locked. The temp would slowly climb past the 190 mark and then the rad would overflow the overflow tank and spill out. I had no problems if I towed sensibly with the OD locked out. Because I have towed this way before without any problem I knew something had to be fixed. First of all my rad cap was still original and wasn't holding ANY pressure and I could not hear my fan clutch kick in. I have owned the truck since new and it had 200,000 KLMS on it and I have never heard the fan clutch come on. So I replaced that at well. I am going to have to wait now to check it until I have the opportunity to tow heavy again on a hot day to check to see if it is fixed. I suspect it will be.

 

In 1999 they took about an inch off the trailing edge of the fan so his will come out much easier. The clutch is a wear part and it is a 1999, I think he should change it regardless and put the cummins 180 degree thermostat in it. All the checking in the world won't make it cooler and there are only a couple of parts that affect it.

 

I just noticed your pictures, on the 1999 leave the shroud in place, don't mess with it.

 

What Haulin said. Also check for debris between the radiator and intercooler. Also another tip. If you think you are getting too warm on a hill and you pull over and let the engine run to cool it down, put your transmission in neutral not park. No pumps are running in the trans in park and you can't cool the fluid unless it's circulating.

 

Just finished the fan clutch install. Bought the wrench set from PepBoys for $20. Had to modify the short one but it did the job. Since I was replacing my T-Stat at the same time, flipping the upper hose out of the way made it easy to remove the fan. I'll have to load the dirtbikes in the back and go up some long grades too see how things work. Still can't figure out why my egt guage stopped working.

 

If this olny happens when you are pulling uphill you should not be using OD, or regular drive, thisis one of the fastest ways to heat up your engine. If you are doing this, you need to put your tranny in a lower gear so that the engine is turning faster, thus pumping the coolant faster and also turning the fan faster. You may already know this, if so forget I said anything. Just thought this might help.

 

I think the new fan clutch will solve the problem.
Mike