If your zig zag is slightly off, it could cause the tension to be slightly off as well. It may be that the zig zag is either slightly stuck. Have you cleaned and lubed your machine recently? Cleaned out the bobbin case area fully?
I always use two contrasting color threads to adjust tension. I've never had issues with black, but I can understand the reasoning behind it.
I think you should concentrate on why your zig zag isn't swinging fully as I would suspect this is your culprit.