Every time I use the trem bar, the low E string seems to go sharp in pitch until I jiggle the bar. Is it possible that the string is binding in the nut or at the bridge and causing it to stick?
Chapstick is fine. I have a stick of Mint flavoured in my guitar case since for ever. I apply it with a needle. 3-n-1 oil is my regular set up sauce. Vaseline, pencil lead, lithium grease, all work well, (though pencil lead leaves dark marks on your nut). Some have used butter or olive oil in a pinch but I'd not recommend those. Some swear by Big Bends Nut Sauce and similar products. Whatever you use, don't use too much and apply it carefully, only where needed. Remember, if you apply too little, you can always add a little more.
FWIW, my low E string still does that. I'm now resorting to reverse psychology - "Go ahead and be a cent or two sharp, see if I care!"