A cold thermostat isn't necessarily a bad thing for engines. Modern, emissions-controlled engines use a hotter thermostat not because an internal combustion engine (in general terms) was designed to run at that temperature, but because emissions are lessened (and gas mileage improved, to a degree) by running hotter. A hotter engine requires less fuel to combust the mixture. So, while a 160 degree stat theoretically allows an engine to make more power, it's not necessarily helpful on a factory computer-controlled vehicle because it's been programmed to operate at the higher temperature. If anything, you'd just see a reduction in fuel efficiency. If the ECM has been modified to take advantage of cooler engine temperatures, then a power gain may be realized. If the computer has not been properly tuned, running a cooler thermostat usually doesn't help (unless the engine doesn't use a computer).
Sorry to get off on a tangent, but to provide some insight to your problem...
My YJ displayed your exact symptoms when I first purchased it. The temperature gauge barely moved from the first notch past 100 degrees, even after an hour of driving (40 degree ambient air temperature -- shouldn't do that). Then, for a while, the needle would constantly show that I was overheating. There were no unusual mechanical problems, aside from a very small radiator leak which was purely by coincidence. I replaced the gauge sending unit and it's been fine ever since. Not sure about the location on the I6 motors, but on my L4 it was in the very back of the engine (head or block, I don't remember) almost up against the firewall. Make sure you've got the new unit in hand, ready to go in, as soon as you pull the old one -- coolant will want to squirt out of the hole because the system is under slight pressure (even when cold; I think it's something related to the fluid dynamics of the cooling system).
You actually CAN check to see if a sending unit is bad by submerging the "sensor" end in a glass of hot or cold water. Attach a multimeter to the terminals and measure resistance. As the temperature changes, the resistance should change in a linear manner. The upper and lower limits of the sensor depend on its design, so I don't know the exact numbers in ohms.
The gauge can also be checked by doing the process in reverse -- power on the ignition and place various resistors across the terminal connectors. The gauge should respond accordingly. I wouldn't do it without knowing what the gauge is designed to operate from, because you might end up pegging the needle and causing damage.