You have some other issue going on, how high on the posts is the bridge ? Is it waaay up there ? If so the neck angle is incorrect and the bridge may be moving a bit on the posts throwing you off. I saw a LP Classic like this years ago for sale, I couldn't believe how high the bridge was on it.
For the record, I have 2 81 LP Customs, an 81 Standard, a 2000s Classic and a 2012 LP Traditional Pro all with stock hardware and NONE of them have tuning problems as long as the strings are stretched, wound correctly, and I ALWAYS tune UP to pitch (meaning if you're sharp you HAVE to tune down below pitch and come back up after giving the string a tug...works EVERY time) or you'll chase your tail. I have a '61 Gibson ES 345, a 6/12 doubleneck (think Jimmy Page), an 81 Explorer, an Ibanez Destroyer and some other Gibson style guitars and NO tuning issues.
Good luck OP...