I don't know about LA, but $250 seems a bit much for the Schwinn. And original tires (and presumably brake pads) aren't a good thing.
I have a daughter who is around your size. A few years ago, I bought an older 24" wheel bicycle for her, and I found out that there are several different sizes of 24" wheel, some of them very hard to find replacements for. (Take a look at Sheldon Brown's page here:
http://sheldonbrown.com/rim-sizing.html.) It would be a good idea to make sure you can find tires and rims in a given 24" size before buying the bicycle.
I was lucky enough to find a very small Trek 820 on Craigslist for $85 that fit her. She's asking for lights and a rear basket now. My wife is around 5' 3" and rides a mountain bike with smooth tires, front and rear baskets, a wide Brooks seat, fenders, and generator lighting.
The rest test of a bicycle is whether it fits you comfortably or not. You might try one of these:
http://losangeles.craigslist.org/lac...664477739.html
http://losangeles.craigslist.org/lgb...636394354.html
I've only bought two bicycles on Craigslist, but it seems that it takes some patience.