cyrokk, also remember that your Les Paul has a shorter neck/scale than the Fender, this makes for a softer feel and it's easier to push the strings out of tune.
Before having jumbo frets filed, definately check your fingering. I have two Strats, the 1961 has small fret wire while the '84 has the jumbo size similar to the Gibsons, it is a far better instrument to play, if only it sounded as good...
With the Epi Paul are you locking the strings when you wind them? eg. on the 6th string- feed it thru, then bend it clockwise and under the string where it goes thru the post. Wind it and pull up on the end piece, this will lock it as it goes round the post. You only need one to one and a half turns. When you have done this give the string a bit of a pull to take up any settling, retune and it should be good, It's hard to explain without a diagram but most people put far too many turns of string on the post, it takes ages to settle. Try locking it and it takes up far quicker.
Clive