I agree with doc.tom that at the heart of this discussion is tuning temperament. I remember watching a player, sadly no longer with us, who before singing would play a G chord, give the B string a little tweak, and then play a D chord. The D chord would sound wrong so the first and second string would need a tweaking before it sounded right. Then he would play the G chord again. Guess what happened next.

I didn't know about tuning temperament back then but I now see that he was tuning his chords so that the 'third' intervals sounded sweet or 'a just temperament third'. Of course the tweaked string was playing a root or a fifth on the next chord and therefor had to be out of tune.

If the OP's problem is not to do with tuning temperaqment then the problem has to do with the instrument. As punkfolkrocker pointed out a high nut can adversely affect intonation, as can an inaccurately placed saddle. Indiscriminate tweaking of the truss rod can throw out the intonation and in certain climates the weather can have a similar effect.

I'd be interested to know if the 'sweetened' tuner corrects the problem.