This may have nothing at all to do with the drop bars, but rather your overall positioning/fit issue on the bike. (when I say "fit" I do not mean "this bike is the right frame size for me", but rather the whole picture which involves many possible tweaks and modifications once an approximate frame size is chosen, and most importantly is DYNAMIC. So fitting perfectly 3 years ago, if it did, doesn't mean it fits correctly for you now).

There are now relaxed fit bikes with drop bars, in addition to several easy modifications that could potentially be made to the Trek that could make a world of difference.

My recommendation would be to go talk to an experienced, respected fitter, either with fixing your Trek or a new bike in mind (but ideally with both). A good solid fitter will be able to see why your Trek is hurting you and either offer suggestions to fix it, or tell you that indeed it is likely not going to work and then recommend bikes that should fit better (drop and flat bar alike). You should also be able to walk out of there with numbers for an "ideal bike" to compare with stock frames. Keeping in mind "ideal" will be different for flat vs. drop due to the different bar widths and reaches.