Engine cooling fans

Those fans will not switch to high speed until the coolant reaches 226 degrees. To determine if there is an actual problem you need to know what the ECM is actually seeing as the coolant temperature. For that you need a scan tool that can read live data (most of the mid range ones will do that now). Or you can use a non contact thermometer on the radiator hose right after the thermostat. The scan tool is a better option if you plan on keeping this car or have others as well.
If you want to check a few things you can check that cooling fan two fuse (15 amp) in the under hood fuse box is good.
That system works by running 12 volts through both fans in series for low speed operation. If the fans are coming on at low speed both fans are okay. Then when the ECM commands high speed operation it sends a ground signal to all three cooling fan relays. During that phase those relays split them back into two separate fans and sends 12 volts to each and they run at high speed. To test the fan control relays you can swap them. The fan one and two relays are the same, if you get low speed fan with them swapped the relays are both okay.
When you select AC does one fan (right side) come on? If yes then the control circuits are working. Then you just need to know the coolant temperature to determine if there is a problem with the Engine Coolant Sensor.

The fan mode card is the controller for the interior blower motor. It uses a pulse wave modulated signal to control the interior blower motor speeds.
All of the cooling fan controls are under the hood. The PCM controls them directly based on the engine coolant temperature system and the AC mode switch position and the pressure switch. Those both combine to tell the ECM that you are turning on the AC and that the AC system has enough pressure to work. If the ECM does not see the actual request for AC and the system activation it will not turn on the fan.

From your description either the mode relay has a bad contact when it switches or there is a bad ground to fan 2. The mode relay is not the same as the other two relays. It has both normally closed and normally open contacts. When it is off the N.C. Contacts allow power to flow to fan 2 from fan 1 and you get low speed on both fans. When it turns on the N.O. Contacts close and it switches the fan 2 relay on and now both fans should be on high speed. If it does not switch the system will stay on low speed.

There is no low speed so that is a concern. I would check for 12 volts at the fan motor and also check the system grounds. You could have a bad fan motor causing the issue. Here is a guide to help you check for power and grounds use the wiring diagram Steve W. gave you in the first post.

https://www.2carpros.com/articles/how-to-check-wiring

Please let us know what you find.

Cheers, Ken

If the second fan is intermittent on low speed then the most likely issue will be a failed fan. It still has a good coil as the other fan is grounded through it for low speed. Check the ground as well as Ken stated. If there is no charge in the AC the fan will not switch on in response to the AC switch but it should still come on with high temperature.