Sounds like a legitimate problem to me. Somebody help this guy!
My solution involves one of these new carpenter's levels - possible brand name "Smart Level". They read out in degrees, and are quite accurate. Check carefully the rib closest to the fuselage, then the rib at the outer edge of the aileron. The Cub is adjusted for equal wash-in on each wing. I have found anything from .6 deg to 1.8 deg, and I personally prefer the lesser wash-in which yields an imaginary increase in speed, at the expense of stall characteristics.
I had your problem, but it was due to unmatched ailerons. One of my ailerons had a defect in the trailing edge, causing a trim-tab effect, and leading me to much fooling around with wash-in prior to discovering the aileron problem.
To detect an aileron problem, level flight, hands off, and watch your ailerons. they should be faired even with the wing trailing edge. Both of them!
Also, be aware that fuel imbalance can affect the heavy wing syndrome. That's what the fire extinguisher is for - you can burn one tank dry, trimming with the extinguisher, and then use the extinguisher to put out the fire after you crash with the other tank full.
By the way, the Smart Level eliminates the need to level the aircraft prior to re-rigging. It gets a more accurate aircraft alignment. And it is not supported by the Piper manual, so be careful about the glowing reports in the log book. As I recall, the Piper procedure is to place a 1/8" block at the end of a 30" spirit level, and that works out to around 1.7 degrees.
bob