while car is moving could be one issue, that is - it jiggles just right to break contact. However, due to the other issue mentioned -- could also be an alternator on it's way out? Get a local car store (autozone, etc) to run a free check. Your reasoning on how the radio works and the constant +12v line is right too.
PS - One other potential issue but this would likely cause greater problems then you are seeing. Some folks have run into a issues with a loose battery tie-down allowing it to bumping up, hitting the hood a little and shorting out that way.