Blown fuse, Heat sensor broken or disconnected, thermostat jammed
I would start at the thermostat. Problems with the radiator and head gaskets are usually presented as visual leaks. But clogged passages in the radiator would also cause overheating issues, so have the cooling system flushed.
I would recomend getting the Haynes manual
It's supposed to run, even with the key off until the radiator is cool. If it never stops you've probably got a bad temp. sensor switch.
Did you check the fuses?
Radiator cap
mine did same, 95 elantra. it was fuse for radio in fuse box. hope that helps.
A radiator cap would not cause a leak! However, if the radiator cap is holding pressure, other weak parts could signal issues in need of attention.
I would suspect a clog in the radiator.
radiator cap
A leaking radiator hose would NOT cause this to happen, you have 2 problems with this vehicle.
Check valve in radiator cap malfunction, replace radiator cap.
Because it would cause the universe to explode