A Les Paul should be a very stable guitar, but always there is the guitar that will give you trouble.
I find the angled headstock on these guitars does well with fewer wraps on the post. I bend an upside-down "Z" into the string, with the bends tight against the post. One wrap goes actually above the free end of the string, the second wrap below it, and I try to stop with those two wraps. Mind you, my guitar would stay in tune for a week with only minor tweaks (this is playing five hours a night, six days a week, plus maybe some practice during the day) long before I started with the fussy nonsense I've just described ...
Something to consider: what are the other guitars, and what is their scale length? If they are very different and you are more used to them, perhaps you may benefit from a bit heavier string on the Les Paul. If, say, you play a Stratocaster and are comfortable on that Strat with a set of strings running .009-.042, then you will balance the string tension and response of the two guitars better if you use something like .010-.048 on your Les Paul. Because we get used to playing our guitars a certain way, you see ... your Strat with very light strings will tolerate a certain touch and technique, and that same touch and technique on a Les Paul with the very same gauge might wander on you.
I hope it is obvious that this suggestion is offered in addition to getting the nut looked at, which is advice you have already received and is definitely the more important thing to do.