I've heard they have so much torque that chainsaw chaps won't stop them.
This is true.
Just like gas saws, electrics come in both cheap homeowner type units and much more expensive, well-made machines.
Stihl's cordless saw would probably be just fine cutting 6-8" diameter wood with occasional forays up to 12" logs.
Check out the MSA160 here:
http://www.stihlusa.com/products/battery-and-electric/
As you can see, the MSA160 is part of a whole group of cordless yard tools. While the cost of buying one tool and the battery/charger may seem high, it begins to make sense if you have need for more than one tool. You can power the saw, blower, hedge-trimmer and weed-wacker all from the same battery/charger combo. You'll notice right away that these tools are nothing like the cheap versions available from Sears/HD/Lowes. I'd invest in the larger battery for lots of saw work. Supposedly the saw can run for 35 minutes per charge. I can do a lot of damage to 12" and under trees in 35 minutes. Even with a small saw.
The rapid charger also is spec'd to pump the battery back up in 25 minutes. Just enough time for lunch.
Note: Stihl is currently offering a $100 rebate if you buy one battery and a tool together.
As far as corded electrics go, I'm still a Stihl fanboy but there are other options. The dealership I used to work for was attached to a nursery/garden center and they sold tons of x-mas trees every year (literally, hundreds). They got tired of the junk saws they were using to trim the trunks and bought a couple MSE140's. That was 6 years ago and they still run those saws.
The MSE140 won't draw anymore current than a typical vacuum cleaner does these days and so long as you aren't running anything else on the circuit, and use a HD extension cord, I think you'll do just fine.
http://www.stihlusa.com/products/battery-and-electric/battery-and-electric-saws/mse140cbq/
I am a woman who is not normally intimidated by things, but gas chainsaws scare the crap out of me.
Hollywood's chainsaw scare flicks probably don't help with that.
I know i'm beginning to sound like a broken record here, but Stihl does make some very "female friendly" gas saws too. Check out the Easy2Start versions of the MS181 and MS211. They're peppy, lightweight, buttery smooth to start and easy to maintain. While I totally get your desire to go electric, if you can handle a gas weed-wacker, then you can most certainly handle one of these.
http://www.stihlusa.com/products/chain-saws/homeowner-saws/ms211cbe/