Mark's suggestion might be very close to the problem. I have owned several Epi LPs and every one has needed the nut filed. They are plastic and need to be adjusted. My tech does it reglularly on new set ups. Even the Elitist needed a slight adjustment to the bone nut. On Visual inspection do the strings sit on the nut, or are they actually sitting down in the grooves? That might be the first place to start. Maybe get a good tech to check out the nut. Or even intstall a bone nut. Remember they are hard to find. A lot of guys in the stores think they know set ups but most will adjust it to what "they" like. Talk to some Musicians who make their living at it and they will know the good techs.
Also be specific when you take the guitar in. Let them know that it sounds terrible in position 1 and 2. My Epis were all great players after a set up.
The Blues is alright!
Guitars: 1968 Gibson SG, 2005 Gibson SG Standard, 2006 Gibson LP Classic Gold top, 2004 Epiphone Elitist LP Custom, 1996 Gibson Les Paul Standard. 2001 Epiphone Sheraton II, 2007 Epiphone G400.
Fender Strats: 1996 Fender 68 Reissue CIJ, 2008 Squier CV 50s, 2009 Squier CV 50s Tele Butterescotch Blonde
Amps: Blues Junior Special edition Jensen in Brown Tolex with Wheat front, 65 Deluxe Reverb reissue,1970 Sonax reverb by Traynor, Avatar Custom 2/12 Cabinet with Eminence Legend V1216 speakers,
2008 DSL100 Marshall Amp , Fender Super Champ XD,Fender Vibro Champ XD
Effects and Pedals: Fulltone Fulldrive II, Fulltone OCD, Fulltone Mini Deja Vibe, Fulltone Fat Boost, Dunlop Crybaby Wah, Boss DS1, Boss DD20 Giga Delay, Boss TU2 tuner, Boss BD2, Ibanez TS9 Tube screamer, Zoom 505. Radial tonebone hot british.