If the fan switch was out, then wouldn't that mean the engine fan would not come on at all

 

Also, what would cause the fan to come on when it first starts to heat up, then when it goes off, it doesen't come on again

 

it is controlled by the PCM which gets its info from the coolant sensor located in the front of your manifold on your passenger side. sensors get sluggish over time, its cheap throw a new one in there and if it doesnt fix it get yourself a wire diagram and run a 25amp FUSED toggle switch in there itll be alot cheaper than replacing the pcm.