I hope you've solved the problem, but from the description of what was happening, I'd be surprised. A tight or binding headset can definitley make bike handling squirrely, but you describe a consistent pull to the same side.
The consistency points not to a handling problem, but an alignment or weight distribution problem. These can be complex with a number of errors combining in ways that reinforce or cancel each other.
If you do conclude that it may be alignment, do not make any changes (except for correcting dish on a wheel) before you know ALL the problems. The carpenter's rule "measure twice, cut once" really applies to frame alignment issues. Measure everything, and get a good sense of what's really happening, then formulate your plan of action.
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