I wouldn't worry that much. Yes, see if you can standby for the earlier flight -- try online checkin 24h in advance; you'll see "standby" as an option, and also check availability at that time. (ORD-STL is not usually overbooked, in my limited experience.)

If you get in and the 7:45 PM is cancelled, or thunderstorms loom -- and keep in mind that even if you connected through IAD, any thunderstorms passing over ORD would most likely also affect STL -- I'd suggest Amtrak. Fare (even walk-up) should be $23, leaves Chicago Union Station at 7 PM and scheduled into St. Louis at 12:30 AM. Take the Blue Line ($2) to Clinton (allow one hour) and walk a block north to get to Union Station. Once you get home, write to UAL to get your ORD-STL leg refunded.

If you are stuck at O'Hare, I've written a short guide to various and sundry options; see the sig.