I'd check the ohm reading of the wiring from the engine block to the temp sensor. I'll be there is a bad connection somewhere that is causing your issue.
Do you know where the ground is for temp sensor? There is something else odd that it does. After the temp gauge reaches between 220-240 degrees, If I shut the vehicle off, wait about a minute, then turn it back on, the temp gauge reads 180 degrees and then slowly climbs back up to 220-240 degrees as the engine runs. It does this consistently. Does this pattern help indicate a particular problem I should look for?
That is odd. It should read HIGHER right after the car is shut off since the coolant will increase in temperature since it's no longer circulating, and then the temp should start to drop immediately once the car is started and the water pump starts to move coolant.
Those sending units are grounded through the head or manifold in which they're threaded into. That's why you
*DO NOT USE* any thread sealant or teflon tape on the threading of those sending units, since it may disrupt the ground and mess up the reading. It's NPT threading: it's designed to seal with just metal-on-metal contact, and if it doesn't, then something is wrong with the threads on your head or manifold. Just in case, it may be prudent to take a wire brush drill attachment and run it along the threads in the head or manifold to clean them up to help support the best ground possible.
You can test your gauge in one of two ways: max resistance and no resistance. First is to test max resistance, and that is simply by inserting your key into the "ON" position and disconnecting the connector off the sending unit. The gauge should bottom out to zero. Then, test no resistance by disconnecting the connector from the sending unit, and then linking a double-ended alligator test clamp wire to the metal female spade on the connector while clamping the other side to a ground somewhere. Key still needs to be in the "ON" position. If your gauge is good, it should then read maxed out (the needle should be pegged all the way at the opposite end of the gauge from the first test, even past the max temp reading - you may have a hard time even seeing the needle anymore). If your gauge reacts as described in both of these tests, it's not your gauge or the wire.