Fan had been running longer than should be for a while lately. Had to jump start three times yesterday, took battery to AutoZone, they replaced with new, put it in and the fan started and would not stop for hours, had to disconnect the battery to stop it.

. Check/replace the Engine Coolant Temperature (ECT) Switch, located on the thermostat cover, not the ECT Sensor.

. Check/replace the radiator fan relay, which is usually located in the under hood fuse/relay box.

There are real risks to jump starting modern-day vehicles. Voltage spikes and, inadvertently, hooking up the cables wrong can cause massive damage; e.g. wiping out the ECM (computer), alternator, and sensors. All of this is to say that you want to get this fixed ASAP, since the risk is greater than you imagined.

If you would like to test the ECT Switch, remove the electrical connector. Using a multimeter, set to ohms, check for continuity across the terminals of the switch with the engine cold. The switch should not have continuity, while the coolant is below 189 degrees F. Start the engine and allow it to reach normal operating temperature. Stop the engine and check for continuity again. The ECT Switch should show continuity, when the coolant temperature reaches 196 to 203 degrees F, and above. If the switch fails to show continuity above this temperature, replace it.

In your situation, you will likely find continuity all the time. Obviously, this will require replacing the switch.