I just spent hours trying to adjust my rear derailleur before I realized that if I'm on the largest cog and I click to shift to the next largest, the shifter is double-clicking and shifting twice but with one louder than normal click. I can click in the other direction perfectly, so if I wanted to hit that second largest cog, I would have to do the loud click and then shift back one. This is why I don't think it's a tension problem, since everything else is perfect except for this.
The shifter is an Ultegra 9 Speed that the previous owner used to replace a Dura Ace one. Everything else on the bike is Dura Ace. Presumably they are compatible, but I don't know. I don't really remember it being like this, but it's possible that I just never noticed.
I'm wondering if maybe the shifter is half broken, and if so, if it's possible to repair it.
Any ideas are appreciated.