Why do air conditioners have thermostats, not humidistats?
Some newer thermostats now offer a humidity setting.  So you can either choose to run your a/c to achieve a specified temperature setting or a specified humidity setting.  (I have an LG window A/C unit that has a humidity setting.)

I live in an area where our humidity level is pretty constant (6%-10%) except for maybe August, so humidity is not a big factor here.  I can see that in some places humidity may have a greater factor.

If a thermostat had a "feels like" setting, which took into account temperature and humidity, that would be ideal.  When the humidity in our area does jump up, my wife often complains she is freezing even though the thermostat is set to the same 72&deg;F it always was.  

If you do replace your thermostat look for one that also includes a "fan circulation mode" that will run every once in a while to move the air around without turning the compressor on.  This can create a dramatic cooling effect without changing the humidity level or wasting electricity.