You should have a second cooling fan that energizes whenever the AC is on. I’ll bet it isn’t energizing.
But that’s too fast a coolant depletion rate to be attributable to just that without also seeking other contributing causes. You may want to perform a pressure test on the cooling system. You may have a leak.
Flow test the system. That will test the radiator for flow and the water pump. You can buy a kit or have it done. Your radiator may be gumped up inside or the pump’s impellars may be eroded.
In addition, I believe this '97 uses the traditional pressure cap on the radiator that releases at 15 or 16 PSI to allow too-hot coolant to blow into and overflow the reservoir bottle, blowing excess onto the ground. Normally it allows expanding coolant to flow into the bottle and draws it back into the engine as the volume contracts when the engine cools, but if it’s overheating it’ll blow too much out and there won’t be enough in the bottle to refill the engine. A new pressure cap is a cheap try.
A thermostat is always a cheap thing to try.
If there’s cloudiness in the coolant, or gump under the cap, you may have a blown headgasket. I’d suggest a chemical test of the coolant for hydrocarbons, but in this case it’s flushing so fast that the test might be misleading. You may want to go straight to a pressure leakdown test of the cylinders. The kit is cheap and easy to use. You simply remove the sparkplugs and put a hose/plug/gage assemblage into each cylinder after turning the crankshaft by hand (via the crank bolt) until both valves are closed (you can see this with the valvecover removed) and pump about 15 psi of air into the cylinder, then monitor the gage and see how much it leaks down. The directions come with the kit. If it’s a distributor-based system, you can also tell when both valves are closed by turning the crank until the distributor rotor is at the cylinder’s pickup point for the spark voltage… it’s easier than removing the valvecover.
The “AC fan speed” is misleading. The only fan speed that controls is the fan pushing the air through the cabin ductwork. That has some minimal influence on the engine system’s ability to dissipate heat, but it isn’t major, and your symptoms are backwards for this to be causative. A “hi” fan speed would extend the time to overheat, your symptoms do it backwards. The fan that complements the engine cooling fan, that fan behind the radiator, only has one speed.
- make sure the secondary cooling fan goes on when the AC is engaged.
- have a flow test done on the cooling system. This will test the radiator and the water pump.
- try changing the thermostat and radiator cap and purge the system of any air, but don’t get optimistic.
- test the system for leakage under pressure.
- test for a headgasket breech.
Post the results. We do care.