The problem with rewinding the string so that it 'sits lower' on the tuning machine post, is that the posts are bevelled, and if the string is not cut with sufficient length to wind the string lower, it can just slip back up the beveled post ... it is a good idea when installing the A string to leave extra lenth to afford that extra winding to the bottom of the post and off the bevelled portion ... temporary fixes for this, if the string is cut too short, is to put a 'zip tie' (or two) around the middle of the bevel, and then wind the string below it .. if that solves the problem, when you change strings, allow for extra winds ...
also, a nut slot can appear fine, but if even cut a hair wide, it can lead to a buzzing as well .. another temp fix here to try to isolate the issue is a thin piece of paper placed in the slot to ever so slightly take up the space around the string in the slot yet still allowing it to slide free ... a couple 'MacGyver' 'fixes' that can help to isolate and determine the issue(s) ...
Great low budget tips from the trenches here! Another I can add, which was laid on me by Eric of Eric's Guitars in LA, a highly respected repair and setup shop, is to use a razor to shave pencil lead into the slots of your nut so as to help lubricate the strings. Eric feels this method works better than commercial lubricants which cost incalculably more.
Sometimes low budget solutions are best -- and some times low budget guitars are best. I love my MIM fretless jazz neck, which now lives on a p/j body. The whole MIM jazz was/is indeed solid.