Well, there you have it in a nutshell. Individual cases of 'irreducible complexity' can be falsified, but ID can't, because, as you say, ' new "unsolved" cases can continually be found'.
It's the old transition fossil paradox. If we find a form C intermediate between A and B, we've replaced one gap (between A and B) with two (between A and C, and C and B). So, Dembski argued, when we found that the flagellum wasn't irreducibly complex because parts of it are homologous to the type three secretory system, then the TTSS is irreducibly complex.
Infinite regress.
I think the claim that ID can't be falsified is aimed at something other than an endless series of challenges along the lines of "You haven't explained this one!" Evolution makes predictions, based on the concept of common descent. Every new fossil must fit into The Tree of Life. If something is found that's obviously out of place (the proverbial Precambrian rabbit), it falsifies the theory. But with ID, literally anything that may be found is yet another wondrous work of the designer. No pattern is required. ID makes no predictions that can ever be falsified, thus (in that sense) ID isn't testable.