I'm not exactly new to guitar (have played mostly acoustic sans pick for a few years), but I'm just now starting to play electric and doing rhythm pieces.

The problem is when I do prolonged strumming, the pick starts to shift in my hand such that I start striking the chords with the side of the pick. Occasionally I'll drop the pick. Obviously, it would seem I'm not holding it tight enough, but when I try to grip the pick tighter it feels unnatural, and it seems like I'm losing "feel" and not producing as nice a sound.

Are there any other things I should consider? Maybe I should head to my guitar shop and by a different kind of pick? Maybe I need to just get used to gripping the pick tighter? Any help is appreciated.
uhmmmm i would have to actually see you to help you in more detail but trying out different picks and different styles of holding them could help you find what you are comfortable with ? cause theres really no one set way to hold a pick its more based on preference( to a point)
pic of how you hold the pic

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well, at least for me, it seems like when i strum harder/faster, i tend to hold the pick tighter and when i strum chords softly, my grip is looser, if that helps. maybe use a lighter gauge pick?
take your picking hand and hold it in front of you, thumb pointing up.
take your index finger and curl it like youd make a fist (LOOSELY)
the last knuckle on your index finger will make a point. put the pick there pointing that way.
place thumb ontop of pick, loosen hand to your liking.
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ok you may want to get a pick with better grip, or may want to check your picking motion.
You may be hitting the strings at a strange angle and it may be turning the pick.
Definitely try a few different types of picks, just so you can see what fits you best.

You need to hold the pick just tight enough that it won't slip at all but not too tight that it feels unnatural.
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Here's a couple pictures of me holding the pick. I think I'm basically doing it the way TK1 suggests. I like to keep my middle, ring and pink finger loose for feel rather than making a fist, but again, I think that's pretty standard.

What constitutes a "strange angle?" I basically strike the strings perpendicularly but sort of strike down at an angle toward the bottom-back of the guitar as you'd expect with natural wrist movement. I'm definitely going to investigate different picks. Any other ideas?