2000 Ford Crown Victoria Overheating at idle and 55-60mph

This is really a multi part issue. The vehicle started to overheat about a month and a half ago. It was assessed that the t-stat may be the cause, so that was replaced. This did not fix the issue. It was found that the fan was not coming on properly. The harness to the fan was replaced and it worked correctly for about 3 weeks. The vehicle then started to overheat again sporadically. It has been determined with a fair degree of confidence that the vehicle is not leaking coolant. The fan is correctly operating, and when the vehicle starts to overheat, it kicks on but still does not reduce the temperature. This also happens at idle. The A/C is cold coming through at idle, but the temp continues to climb.

The vehicle on a recent road test went through 6 separate stages. First started running normal leading up to a nearby highway (30-40mph, highway 50-60mph). Upon entrance to the highway, the vehicles temp started to climb. It was taken off the highway back to 30-40mph speeds, and it started to cool almost immediately following the speed reduction.

Brought back onto the highway for ~2miles, and started to overheat again. Again was taken to a 30-40mph road and again started to fall in temp. This was then followed by the last few turns back to the driveway, specifically at a stop sign, wherein the vehicle again started to overheat. It continued to do so even when it was parked, as it has done before. The temp continued to climb despite the fan being on and coolant level well within the green.

No apparent oil in the coolant, or coolant in the oil. No odd smoke from tailpipe.
The engine does run a little rough when it is overheating.
Have replaced 3 coil packs a year ago.
Have previously replaced 2 power steering pumps.

Current known possible causes:
Water Pump not functioning properly
Radiator plugged/clogged
Head Gasket damaged
Thermostat bad
Air pocket in coolant system (plausible, there is no bleeder valve, possibly could have formed from replacing t-stat previously)

Additional diagnostic information:
-No fuses blown
-Coolant level is in the middle of the fill lines when cool, seems to run high when hot
-After last overheat, checked when cool and lower hose is empty, while upper is full of fluid. I am not aware if that is diagnostically relevant.