Check working bikes at 2434 S western, show up early as the good stuff goes first. The hours and days they’re open are weird, something like Wednesday and Saturday 10am - 4 pm. They’re a non profit that rebuilds and sells bikes, so the prices are good.
Go used, you’ll get a better frame and components for that price. Look for good steel and solid components, like some old Shimano Ultegra (or 600). They’ll last a long time.