Never played one I dug until about a week ago. Went in to a local shop for something and my regular guy was busy. He handed me a 2009 Singlecut with a Wide Fat neck and pointed me at a Mesa Roadster and said he'd be with me in a few.
Hour later I was in love and had a small crowd and another guy jamming along with me behind me. There were some things I didn't like compared to a LP, among them the finish and the weight. The finish feels kind of plasticky, and it was a pretty light guitar; around 7-8 lbs max. I did really like the neck, it was perfect for me; also loved the tailpiece-without the strings arcing up over the saddles like on an LP they were pretty close to the pups; just felt really natural to play.
I know you play LPs and dig em; try a Singlecut if you can find one. I loved that guitar, man. Seriously, it was one of the best I've ever played. My next purchase is an amp because I alternate between the two, and I've just bought/had built a custom guitar, but after the amp I'm on the hunt for an 09 Singlecut.
ADDED: bro, I just grabbed whatever he threw me to play to have something to do while I was waiting. I realized that one of the reasons I liked it was because every other time I'd play one I'd think to myself "this doesn't sound like a LP."
It's NOT going to sound like a LP, except that they're both guitars. They're lighter and thinner, for two things. Keep it out of your head that it doesn't sound like you think it should, and go play the crap out of every one you can find. It's taken me years to connect with one.
Good luck!