I know this problem well - and it is a pain in the arse as it makes the 3rd in many chords sound sharp and evrything sounds a mess.
If you check the intonation at the 12th fret and it seems ok yet the sharpness remains on the first 5+ frets then I would recommend looking into an Earvana nut as one way of tackling the problem
www.earvana.comI am one of the main stockists for these in the UK so am pretty conversant with them
the drop in shelf nut is the simplest to use
There are also other benefits to putting one on - especially on a Les Paul or SG guitar
Makes the guitar seem more alive and balanced IMO
I fit them as standard on my custom builts these days
PM me or give me a call if you want more info