Yes, you should try a surge protector. Preferably one that also protects phone lines. I have seen strange problems with Static Electricity And/or transient voltages induced on appliances specifically computers and network systems. You may have one or the other feeding back through your receptacle. Properly grounded systems will usually eliminate static electricty problems. As for transient currents, they are caused by current flowing through a wire that passes close to another wire. If the phone system wiring is close to the new breaker box or the wire that feeds the air unit it may be inducing a voltage and current into your phone line. This could cause the base unit to do strange things.

The old grounding system for your house may not be adequate for your new panel. If you had an old service of less than 200 amps. you may need a larger ground rod and electrode conductor.

These are just a few guesses at what may be wrong with your phone. Stuff like this is hard to diagnose without being there.