Would it do harm driving on an broken transmission solenoid

Hello, if the solenoid is the literal cause of failure, then yes - you can cause problems further mechanically by not repairing it. The thing is -- you should be trying to get this resolved as soon as possible to avoid the concern getting worse. Also, are you over 60,000 miles? There is powertrain warranty for 60,000 miles or 5 years on this vehicle. Seek guidance at your local Chevrolet dealership.