| Nope, a truss rod will have very little, if any noticeable effect on your intonation (unless the neck is grossly backbowed or upbowed).
Is the E string sharp all the way up and down the fretboard? Or only in a few specific places?
What quality bass tuner are you using? Really cheap tuners can sometimes read the intonation incorrectly.
Are you sure that the string is contacting the saddle on the right point, and not a little farther forward than it's supposed to?
Did your last set of strings have proper intonation for the dropped D?
The reason you have to move the saddles back to get proper intonation is that the strings stretch a little bit when you fret them, raising the pitch slightly. Moving the saddles back effectively lowers the pitch enough to compensate for the stretch. A string that is tuned down to D from a standard E will stretch more and thus require more compensation, which your bridge may not be able to give (shown by the fact that you have the saddle adjusted as far back as it will go). You might be able to improve your situation by getting a heavier E string. The heavier string will require more tension, and thus will stretch less, requiring less compensation.
Also, lower action will require less compensation as the strings will have to travel a shorter distance to the fretboard when fretted, thus stretching less.
Get back to me on the questions I asked above and I can probably help a little more. | 
