2003 Jeep Grand Cherokee Overheating When Idling?

Hi, I'm a 32 year old female so please bear with me as I am probably not as
mechanically inclined as most guys! I have a 2003 Jeep Grand Cherokee Lerado. It
does not have a towing package. Last summer it started overheating while idling,
never while driving, only when I'm at a standstill or in traffic. My thermostat was
changed, didn't fix problem. I checked all the fuses, they all appear to be good. My air
conditioning works fine. My cooling fan turns by hand, with no resistance, but it never
kicks on at any temperature. I changed my fan relay and harness myself, it didn't fix
the problem. My engine runs about 183-185 degrees steadily while driving, then climbs
all the way to the redline (260+) within minutes if I'm parked or idling. It has overheated
to the point of coolant bubbling over and onto the ground once, before I was aware of
the issue. I added more coolant and it ran fine after it cooled down. I do not know how
to "jump" my fan to see if it works? I would try to test it if i knew how. I cannot really
afford to take it to a shop right now sadly. Is the next logical step to try replace the
whole cooling fan? Also, when I replaced the cooling fan relay, I did it the easy route
by removing the passenger headlight and accessing the relay through there, and there
was already a hole cut out with tape over it exposing the old relay so I'm assuming this
isn't the first time someone has changed it. I am not the first owner, but the cooling fan
itself appears stock? I am just so frustrated because it's hard to drive without having
to worry about it overheating in traffic. I appreciate any suggestions! Thank you for
your time.