So, it hasn't been until the last few years that I've ever considered myself a "musician," and in all honesty, when it comes to playing music my primary interest is in songwriting.

As such, my main instruments are piano and acoustic guitar. I also play bass in a three-piece band with my friends. I've been playing an acoustic guitar for close to 10 years now, and I'm pretty much where I want to be with that instrument. (I mean, I can always get better, but I'm able to write songs and play pretty much most things I want with some practice.)

However, in my heart I want to rock, and a lot of these songs that I write I usually intended to turn into full-band arrangements. So I bought and electric guitar and I've been jamming with my brother (who plays drums), but I am having trouble transferring my acoustic skills -- especially when it comes to jamming or improvising.
I tend to rely on just playing barre or power chords and kind of running through the same progressions and rhythms - OR I end up just playing the electric like an acoustic when I'm really stuck, and with distortion a lot of those open chords sound mucky (as you know). I try to vary it by inverting power chords or playing fourths or more lead-type stuff, but I'm still pretty unhappy with it. It also hurts my left hand a lot more and a lot quicker and my playing is MUUUUCH sloppier.

My question is what is the best way to adapt to playing electric guitar? What techniques can I use to improve and become better at jamming/improvising (aside from scales, which I already do)? Mostly, how can I adapt to fit the electric guitar better?

And just to start a general discussion on the differences between the two instruments.
If you're just starting on the electric guitar, turn that distortion off dude. You should ensure that you're using good technique on a clean setting first.
And no, Guitar Hero will not help. Even on expert. Really.
Since you can play chords and what not on the acoustic guitar, warm up with a few chord progressions on guitar. Try some chromatic exercises as well; they'll allow you to help clean up your technique a bit on the electric.

As Alan said, play on a clean channel setting. You want to make sure that your technique is correct. Otherwise, the distortion will cover up your mistakes and you won't improve.
Skip the username, call me Billy
Your hands hurt on an electric? That's a bit weird as I'd imagine the opposite. The action on electric guitars are usually lower, so it should be much easier playing.

Anyways, just like the first post says, I'd suggest you play on clean setting for a while so you can hear your mistakes. Learn some scales, pentatonic is the best one for rocking out imo, and just practice those in all five shapes. Seek help on YouTube for different licks and ways of enhancing your pentatonic (i.e. mixing pentatonic minor and major), and soon enough, you'll be sounding like a rock musician.

For me personally, scales have completely put me over the edge in my musical and improvisational ability. Once you get that down, you'll just be able to nail down a solo on-the-fly to any pentatonic chord progression. Hope that helps?
Don't start with pentatonic scales or pentatonic chord progressions(a ii-IV-V-I ?)

Practice your major and minor scales from multiple octaves. The pentatonic scales are already in these scales shapes and you can easily see and feel them.

Study some rock riffs and see how they match up to the scale, you might find some chromatic thrown in for color.

You're only as good as what you know, so study.