The way i understand it, if there's something catching the string at the nut (particularly if its a plastic nut) and you use the trem, it's going to mess up the string pitch. A simple fixer-upper is to get some lip balm, put a bit on the end of a toothpick and smear the wedges where the strings sit, just so there's a light coating. This way, the strings won't get caught as much and can slide like they're supposed to on a non-locking guitar.