Well, I'm not in Omaha, but down the road a piece in Lincoln. Maybe I can help a little, though.
First, I know what you mean about Omaha drivers. I HATE driving in Omaha. As with any city, though, there are streets, and there are streets. Often, about a block or two over from a major arterial there is a perfectly good biking street that isn't cool, or fast, or something enough for cars. Get a city map and plot your route. After all, with every car in the universe on I-80, there have to be some open roads elsewhere, right?
I think you have bike trails too? Try them and see if you can work them into your route. A dry run on a weekend might be a good idea to refine things. Things just look different from a bike seat.
I'm sorry if this isn't site-specific enough, but that's what I'd do in your cleats, er, shoes.
Give big space to the festive dog that make sport in the roadway. Avoid entanglement with your wheel spoke.
(Sign in Japan)
1978 Raleigh Gran Prix
2003 EZ Sport AX