I would replace the capacitor if I was you, continuing to do that to your fan can burn out the motor eventually. Note: I cannot edit what I posted a couple posts above (pre houzz merge), what I said about the capacitor was wrong. The capacitor charges constantly when supplied power, it just discharges when the fan is turned on, giving the boost needed to make the fan move from stand still. After that, the fan motor keeps the fan moving. It wouldnt hurt to give it a minute to charge before using after replacing with new. In reality it probably only takes like 10 to 15 seconds. That is what I was going for with that comment. Probably the time walking from the panel to the switch is long enough.

The bulb in this socket draws enough current that the spurious voltages will not occur. I assume that there are two black/white pairs which come from the breaker box. It's easy to see which black is which. First, connect the white on this socket to the twisted whites and touch the black on the test light to each of the blacks and see if any of the blacks are hot. Once you have confirmed that at least one white/black pair is powered, turn off that breaker and make these temporary connections more secure (not just touching live wires). Turn the breaker back on and then turn on and off the likely wall switches. If none of the switches make the test light go off, then you have found a source of power. Repeat the above for all the black wires and see if any of them are switched. Finally, and I'm not sure about the following test, but if one of the black wires never shows power, it must go to another light fixture or receptacle. Connect your test light between the black power source and this unknown black wire. If your test light glows, then that black wire goes to something which draws current (perhaps a ceiling light with an incandescent bulb in it), and you can figure out what that is by observation. If it doesn't glow, then take another test light, put a regular plug on it, and plug it into each of the likely receptacles. If this second test light glows, then the two lights are in series and you know where this unknown wire goes and you can nut it to the power source wire (or to the switched wire, if you want that receptacle switched). One thing I don't know is how to trace out the white wires. If each of them act as a reliable neutral, then maybe it doesn't matter. If it does matter, I don't know how to identify which is which. Sorry for this long response; I hope I'm clear.

I have had similar problems with ceiling fans. For my first home, I bought cheap fans with pull chains and light kits. I wired them with two separate lines funning from two wall switches. I control the direction and speed with the chains and fan-mounted slide switches, but normally I just use the wall switches. The line going to the light has a dimmer switch; I haven't tried a speed control wall switch because the pull chains work fine for the few times I change the speed. So, for my new house, I had the electrician run three wire (black, white, red) plus ground from the wall switch box to the ceiling fan box, with the plan that I was going to get and install the fans later. My first fan was a fairly high end fan with a DC motor and a remote control. The connection in the fan to the light was inaccessible, so I just used one circuit and the remote control - bummer. It turns out that that's not too bad - I turn the fan/light combination on and off with one switch, and use the remote only when I want to change something. I went to Lowes (where I bought the fan) several times and complained, I finally connected with a salesman who knew what he was talking about and he said that if I want to hard-wire the light and fan with two separate circuits, I have to get the cheaper "pull chain" style. A knowledgeable salesman from Home Depot confirmed this. I then installed two fans in two separate bedrooms. One was a fan identical to the one I mentioned above and the other was a sale pull-chain fans. The both work fine, but the pull chain fan is much noisier than the one with the DC motor. The pull chain fan cost about $45 and was on sale for $10, while the one with the DC motor was $179.

There are two hardware components. the receiver with antenna was installed at the ceiling location between the house box and fan. The second component was the wall control which is what I displayed. This should swap with the existing switch, however the wiring is not the same configuration. old existing switch was single gang, dual slider with separate wires for fan, light, and hot. New switch has only two wires for fan and hot. attaching pictures of the instructions for further clarity.