I recently had a very similar situation as the OP in this thread and I posted a related question at a different link.
My "cheap" low end electric dryer (rated for 220/240VAC) stopped working after several years. After checking all the obvious things like the circuit breaker, switches and controls inside the dryer, and the several thermal fuses inside it also, I was unable to determine the cause of failure as all of those checked out correctly. I did not see any scorched or burnt wiring anywhere inside either.
I then checked the voltage at the dryer socket as the OP did and noticed it was not correct. I was getting 120Vac from one line to neutral, but only around 60-70Vac from the other line to neutral. I then called a local electrician who said it could be a bad breaker. Even though I have some electromechanical skill, I decided to let them replace the faulty breaker - which was the cause of the faulty readings at the socket all along! I plan to find out what exactly died inside the old dryer when I take it all apart eventually...
Please see the pictures below for the damage that occurred to the breaker. Inside the contact bus in the panel, there was several globs of "molten" metal that was where things were arcing, etc. I have no idea how long this breaker has been this bad as I noticed no obvious symptoms running the dryer - except for the day it would not start at all. This was certainly a fire hazard....
The electrician said that the panel was so old (which is true) and several more contacts were very corroded that everything will need to be replaced and a dehumidifier also may need to be installed. I will be getting that done soon. I wanted to post these pictures to let other people know that "faulty" readings at the socket could be caused by issues such as this with the breaker.

