hello jsteven2001
Sounds like you need to replace your themostat, because the fans are not supposed to run for more than 10 or 15 seconds after the car is turned off if at all.As far as the needle on the temp gage, does it look like it's stuck below the stop peg on the gage.
This happens usually when somebody has been in the dash doing other work, If not you may need to replace the temperature sending unit.
Chevy Technician
ASE certified,expired. 30 years experiance self employed semi-retired.