I could help you if i knew your set-up with the AC and furnace. I am assuming you have a programmable thermostat that is controlling both your heating and cooling unit and has a set temp that determines when each device would turn on or off that someone had to program i’m assuming, the problem may be with the programmed temperature settings themselves.
You didn’t give enough information for me to accurately help you, i’m assuming the AC and furnace are wired into the same thermostat since you mentioned that the AC stopped working when the new furnace was installed, and some people i know have this set-up in their own homes. If i’m wrong than is the AC hard-wired or on a plug going into an outlet and is there a switch that controls the outlet? I don’t have enough info on your set-up. I do know if your temp setting is to low it will not supply power to your switch so check your temp settings first.
I have to leave for work in an hour but will try to log back on later tonight to try to help you. P.S make sure your power is off before testing for continuity!