Radiator fan problems or is it a sensor or something worse???

My 2008 Mini convertible had recently & only sporadically, begun to have the
radiator fan stay on after turning off the car. (BTW: not hot outside.) Since the
dealer is out of town, and since the problem was only sporadic, I have putting
off making the long drive. Now, a couple of days ago, I just happened to look
down at my dash to find that my car’s temperature had red-lined. Thankfully,
the fan then did turn on and the engine cooled down. Now, I’m afraid to make
the long drive to the dealer for repairs. Concerned that a general foreign car
repair shop won’t use the proper parts, etc. So, I’m trying to trouble shoot
what possibly is wrong. Maybe coolant is low? Since I drive very little - and
because of Covid - I skipped my maintenance trip in 2020. Plus the car is now
almost 13 years old. It could be my thermostat, the fan iitself or even the
radiator - or it could simply be a sensor. I’m a newly separated lady in my 60s
and I honestly don’t know what to do!!