The words are similar, but not close enough to be synonyms. To plumb something means:
to try to understand or succeed in understanding something mysterious
This is a very literary usage. Further, if you plumb the depths of something you are saying that:
to be or to experience an extreme example of something unpleasant
I think you're mixing phrasal verbs: to track something down means to go look for something. To plumb something means that you are trying to understand it. But you shouldn't use plumb something down because that is just a mix of the two. So your second sentence should be:
I've plumbed the problem and found something blocking our way.
To plumb something is not likely to be as fully understood as "track". As the dictionary says, it is more of a literary usage. Stick to tracking something down--it will be more understood, which in a programming context is important.