The fan will help you while you are standing in front of it. Otherwise, not so much. If you have a window A/C, it is fairly simple to check out. When you remove the filter for cleaning you are looking at the evaporator coil (lots of shiny aluminum fins that can give you lots of nasty shallow cuts). Is the evaporator coil clean? If not, a clean hairbrush or similar tool can be used to comb the fins. Second, is the unit producing plenty of water to the outside drain? Are the evap coils wet? If yes, then you probably don’t have a freon problem. Last, if it is accessible, is there a whole lot of hot air coming out of the back of the A/C (the part that hangs outside the window? If it’s really hot but there’s not a great deal of it, then the problem is a clogged condenser coil. This is much mmore difficult to clean, since it involves removing the unit from the window and or disassembling it from the outside, but it is one of the most common problems with window units. If there isn’t enough air moving over the condenser, then the unit cannot reject the heat it is drawing in from your home, hence, not much cooling takes place. It also causes the unit to consume more electricity than normal and run longer.
A fan can help distribute air through the room more effectively, so use one. Ceiling fans also help control the heat distribution and offer some cooling while you are under it, but only use electricity and add heat to the room when no one is there. Turn them off when you are not home.