There's a solution for dealing with wood screws which have broken off below the surface of the wood. If you don't want to risk damaging the wood by digging it out, try driving the screw deeper into the wood with a nail set. Then just fill the hole with a wood filler, and drive a new screw next to the broken one.
Don’t set your thermostat at a colder setting than normal when you turn on your air conditioner. It will not cool your home any faster and could result in excessive cooling and unnecessary expense.