| Great that you've narrowed it down, it makes sure you only end up swapping the bit that's causing the problem.
Now just before you buy a new solenoid, it might be worth undoing and cleaning the two terminals on the solenoid itself and seeing if that has any effect - if not then it's fairly obvious that the sol is fooked.
You'd be amazed what high resistance due to poor connections can do and the smoke you originally mentioned suggests something was getting rather hot.
At worst, a replacement solenoid should not set you back too much, I've seen them on ebay for not much more than £20. | 
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