OP, many songs are played half a step down (whether that is 435 hz or not I don't know) so that's why you are in tune with the songs but not with your Korg. Tuning your strings to a song or to standard 440 or to a slightly out-of-tune piano is different than intonation. Setting your intonation correctly will make all of your fretted notes sound "in tune" with whatever tuning you set your strings to.
Basically, there are 'imaginary' points on each string that represent the 'in tune' note that you play when you fret the string. By setting the intonation you are matching these 'imaginary points' on each string to the actual 'real' frets on your neck - so that when you play a note, it sounds in tune.
This is performed by adjusting each saddle on your bridge either closer to or further away from the nut. To start, tune your "E" string to standard E. Then fret the string at the 12th fret. Check your tuner. If the note is sharp, that means that the vibrating portion of the string (between the 12th fret and the bridge saddle) is too short. Therefore, move the saddle away from the nut using whatever adjustment method your bridge has.
If the note is flat, the vibrating portion of the string is to long - move the saddle closer to the nut. Only move in small increments; if it's held in place by a long screw, just a couple turns. You'll get the feel for this after doing a couple of them.
Once adjusted, re-check the open tuning of the string (as it may change) and then check the notes past the 12th fret closer to the bridge. If set correctly, ALL frets on your neck - from 1 to 20+ should be perfectly in tune when fretted. If frets 1-3 are a little sharp, this means that your nut slots are a little too high - a guitar tech should be able to file them down for you.
So there it is; once you learn this you'll save hundreds of $$ by not having to take your axe to a shop to have strings put on or adjusted.
Next week, neck class 102 - how to set your action and truss rod...
Rock on mate...