Had issues with the new LP I bought recently too, with those horrendous black-plastic-covered pickups. Anyway, in my case was a nut issue, had to lube it with graphite alot otherwise it would catch a string while bending and it'd go out of tune easily. I eventually contacted a local guitar builder, and he made me a bone nut which I then fitted on it, top-loaded it with elixir strings, and never had any issues with tuning stability ever since. The guitar in question also had issues with frets and I had to polish them and level them properly and this was brand new guitar too, not used one.
So if I were you, I'd try top-loading her and lubing up the nut (you also might the tunomatic bridge as well) and see if it helps.