A 31" inseam on 5'5" frame is pretty unusual. In general, women have shorter legs than men, and most 5'5" to 5'7" men would be dragging trou legs with a 31" inseam. With road bikes, the fit is going to be more important than that Sports.
At 5'6" and with a 30.5" inseam, I ride a 56. Charts put me at a 54-56. I'm guessing you'd be comfortable at 52-54cm. Given the measurements you sent, you've probably got a shorter torso, so a top tube of 52cm may be all you can handle, with a medium to short stem, too.
$100-$200 would put you looking at yard sales, as the CL bikes in your area can get pricey. Schwinns carry different brand recog in different areas. I agree with Poguemahone, a Greenville Schwinn could well be underpriced for it's value, especially an aluminum one (but make sure you get the seatpost on one of those). A Centurion Lemans, Nishiki Sport, Panasonic Sport, Univega, Shogun, Miyata, almost any number of mid-80's Japanese bikes would fit the bill. Best idea is to find some examples and float them via PM to folks who may know, especially in your area.
I find Technium's pretty reasonable, but probably wouldn't pay over $150 for a really nice one. I fully agree with Poguemahone on Ross bikes. Very cheap here.
The market here is flooded right now with triathlon bikes and folks selling that bike to get ready for school. Not a bad time to buy at all. You may also want to check with the local shops on trade-ins, tell them what to look for, so they don't have to take one in, rehab it, and charge more than the bike's worth.