Ballast resistor problem kind of presents itself opposite of what you are experiencing. Starts until you release key, instead of what you describe as firing up as soon as you release key.
It can work either way.
On a four wire ballast it can go bad on one side and act completely the opposite.
Since there is voltage in both start and run positions but low on the start side it's most likely the ballast.
They do go bad inside and give low voltage, and the trick is to get the right one to replace it with.
