There is no easy and quick way to find the leakage. This is a trouble that requires old-fashioned and tedious labor to locate.
From your description, though, it sounds like it's not wet cable (which would present a steady reading above zero, and not eventually decay) but rather a device, like a load coil, NID with left-in wiring, or a bridged terminal with a left-in drop. The "jumping" you describe could be a wet pair, but I'm leaning towards some device that's doing it. On the other hand, the points that you measure indicate a balanced and open pair, which is a good start.
If you can meet the maintenance splicer in the field, have him open the drop wire, which seems to be one of the things you have not been able to do, as a contractor. I have seen seemingly perfect drop wire exhibit weird problems.
Once the entire pair has been isolated, ask him to go along the route, following the information on the cable plats (diagrams) and open the pair at a mid-point that's convenient.
Then with the pair open at both ends, do the test again. This will eliminate half of the OSP. Then ask him to do it again, cutting the bad half in half again, and so on, until he finds a bad coil, or a branch that is not on the plats, or a left-in drop wire at a bridged terminal. It might even be a deteriorating phenolic faceplate on an old pole- or ground-mounted terminal / cross-box. I've had them act as a resistor, capacitor and even, in one case, a diode, allowing leakage in one direction, but not the other. A quick spray with "instant switchman" and some WD-40 fixed it temporarily. The splicer they send needs to be able to think of all the possibilities.
Can you take a look with a plain old "brown" meter, rather than the Sidekick? I used to love to get involved with these troubles, and found that the old KS meter sometimes gave me a better result than my Sidekick.
