Rule number one - only use the absolute minimum amount of lubricant necessary. Too much could dull your strings quickly, especially the wound strings as the lube could penetrate into the windings!
Whatever lube you use (nut sauce or graphite or chapstick or mineral oil or whatever) make sure you periodically clean the nut slots and saddles (maybe every second or third string change). Old sauce could serve as a 'sponge' to acids and oils from your hands, and could also dull the clean, firm contact the string should be making. It will also eventually get sticky (which is not what we want).
I would try cleaning the slots and saddles and first just rubbing a dark pencil on the saddles and nut slot and see how that works. Why? because graphite is a dry lubricant and causes less problems than wet lubricants do (provided we can live with a little pencil mark on the nut).
Strings can also wear notches or indentations into the saddles, which might possibly be causing this. (It can also lead to more string breakage). I like to use a small pieces of 450 and 1200 grit sandpaper to get rid of the notches. Really bad notches may need a small file first, then the sandpaper.
