Here's an easy way to check if the solenoid is dropping out. The solenoid, once activated, is only held on with a latch. Any interruption of power to that latch will cause the solenoid to disengage and stay disengaged. It will only re-engage by turning off the key and going to Start again. So if the engine quits, leave the key on, open the hood, and look at the solenoid. If it is still engaged then it is not the problem. If it has dropped back to the disengaged position, then there is your problem. Actual cause could be loss of electrical power or an internal problem with the solenoid. This test should either clear the solenoid as your problem, or cause further tests to determine why it dropped out.