Do not try to add water or coolant while the engine is running or hot. You will only risk hurting you and the car. You only add fluid in this way after a system flush wherein you leave the cap off after refill and keep filling as the thermostat opens.

The fact that your aux fan goes on tells me your temp gauge is probably accurate (the first question to ask in my opinion,) The three most likely components related to cooling are the thermostat, viscous fan clutch, and radiator. Thermostats sometimes fail, radiators get clogged, and fan clutches wear out. There are threads here (use the search tool on the website) to learn more about replacement of each. For a different car, 1984 380sl 107 series car, I wrote DIY notes on all of this. See threads started by me, click on ericgr.