It sounds like you are having issues with your electrical panel or wiring. either a dropped neutral or perhaps a disconnected neutral on either the dryer circuit or the furnace circuit. or a bad connection on Lead 1 or 2 of the dryer or furnace. the dryer most likely.
When the electrician came out did they find you have true 240 volts between lead 1 and 2 coming to the dryer and then did they check for 120 v between each lead and neutral?
I suspect an issue with lead 1 to the dryer which powers the control panel and starts the motor. Lead 2 is only used to power the heating element.