I see that this question is still getting some attention after all this time so I thought I'd come back and share what I found to be the problem.
The question itself was predicated on my assumption that since I had recently replaced the transformer it was unlikely to be a transformer issue. As it turns out that was a poor assumption.
After some time we ended up replacing this set of lights with another of a similar kind. As I was doing to work to replace them I noticed that I'd never trimmed the excess LV wire from the end of the run (these are the type with bare braided wire that carries the LV current as well as supports the fixtures) and at some point, probably when someone was dusting up there, the 2 sides of the wire got stacked on top of each other creating a big old short. I suspect at this point that that short burned out the transformer much more quickly then I considered likely and the now-faulty transformer caused the odd behavior.
In any case the replacement system (with its nicely trimmed LV conductors) has been working flawlessly for quite a while now.