Generally, there are not 2 fans. Instead, there are electrically controlled dampers that open and close based on signals from a control unit. The damper for your upstairs vents is probably jammed.
The control unit for the dampers is usually mounted on the wall near the air handler, or directly on the air handler. There should be 4 to 6 wires coming out of the controller: one for each thermostat, one for each damper, and possibly one or two more going to a power supply or to the air handler. They're probably labeled if you open the cover. Follow the cables to find the dampers.
The dampers that I have in my house have a cover that you can pop off to get at the damper and move it by hand.
First, I feel your pain. I'm in Houston at my girlfriend's place; we live in a single family house, and the A/C is running all day even with the thermostat at 84 degrees. I work from home, so I can't let it go up higher than that. Our outside thermometer recorded 106 degrees today.
The things that we've added to the house to help keep cool, besides new insulation and having the A/C unit checked, include a bunch of things. First, a little theory. Your apartment heats up because of the sun heating the building primarily, and secondarily because of infiltration from the outside of warm air.
I'm going to just assume that your landlord isn't willing to replace your windows and doors with low-E dual-paned units. I've done this to my house in Bryan, TX and it literally cut my utility bill in half.
The first thing we did with her place, where the landlord is also not willing to rip out a ton of windows and replace them, was to get blinds with reflective shades in all of the windows, and external solar shades or screens where possible. Remember: Black stuff is OK on the outside of the structure, because black absorbs heat (and you don't care about the heat if it's on the outside) -- and white things on the inside of the structure, because you want the heat to not be absorbed and to just be reflected back out.
The second thing we did was to get a better thermostat for the house. The old one was a manual one with mercury in it; the new one has a digital timer and a bunch of other features that include breaks for things like the compressor coils to defrost.
The third thing we did was start replacing the A/C filter frequently... at least once a month.
It might help to suggest more if we knew what kind of A/C unit you have (is it one of those high-rise below-window units, or a 'real' one with a compressor and air exchanger in a closet or attic?), and what kind of structure (concrete or wood frame) you're in.
Best Answer
If the lights are dimming several times while the unit is running, the only thing you can check is listen to the outdoor unit to determine if the compressor is unexpectedly starting and stopping when it should be running continuously. Set the thermostat to a low setting and listen outside to see if you can hear the compressor starting and stopping at about the same interval as you notice the lights dim.
If abnormal starts and stops are occurring, a competent technician is needed to evaluate and repair the problem. Whether or not the compressor is cycling unexpectedly, installation of an appropriately sized 3 wire hard start kit can sometimes alleviate lights dimming on startup due to the electric service being marginally small for the unit.
You may want to try a better technician than the one who advised an entirely new system and then replaced a leaking capacitor. Any technician worth his salt would use a capacitance meter and not an amp meter to legitimately investigate this problem. Also, unless your 6 year old system was installed absolutely horridly, it should not need replacement so soon.