I was happy to find this thread, as my Ultegra shifter is doing the same thing. It took me a while to realise that I also had to reach around and stabilize the big lever while clicking the small one, then it works flawlessly. However this is a major hassle and distraction when trying to concentrate and stay aero.
I looked inside and the screw was actually gone, so I went to the hardware store and bought a rough approximation.
Now the levers feel "stable", not mushy like before , but it still won't shift into higher gears and i still need to hold the two levers in order to shift.
I have even learned a way to do this with two fingers while still holding the hoods.
Interested to see how the warranty thing goes as this bike is one and a half years, but my bike shop is in another country.
But, what IS the actual defect?