Sounds like the string is slipping around the old tuning post to me too, time to take a look at how you are stringing that tele up, if you want to do a little test, tune it up and mark the string with a marking pencil right at the place the last wind comes off the post, play it until you lose tuning and take a look at where the mark is, if it has moved any then you have a slippage problem, another thing you might want to check for is slack storage, tele's have those string trees's that will some times hang the string up, when you tune always go from low to high, ie.. tune the e string up from D# to E not from F down to E, and when you tune up grab the string at about the 12th fret and pull it out and away from the fretboard, this pulls that stored slack out of the pinch point wherever it may be, I take this is mainly a problem you are having with the unwound strings?......................