I'm also quite sure that the peg needs a refit. But until you get a chance to have that done, a quick fix could be to add some soft blackboard chalk to the peg dope. And obviously make sure that the string is rolled up against the pegbox wall towards the peghead.
It does sound like something is wrong with the fit; either there is a worn ridge on the peg, something out of round, or some taper mismatch. Refit sounds like the right thing.