Specifically to your question, I have only been in the UA clubs in DIA several years ago and don't remember whether or not they have showers. My guess is that question has been raised on the UA board. I don't know their rules on a day pass if you are not flying UA (or what time they close). BA has or had a club close to the F9 gates, so maybe check into that as well.

But you may not need to find a shower in DIA. Once again, Webflyer is your friend. Check out the flight schedules and I think you will find several options leaving at 6PM or later which should get you to Philly that night. http://www.webflyer.com/travel/flight_schedules/ (obviously don't know what the availability is on the days you need to travel).

Also, since OMA is pretty close to the city and if you have luggage to check, you might consider running it out there at lunch and checking in then.

If you are not familar with Omaha and are renting a car, ask about the locations of gas stations when you pick up the car, in case you have time to refill it. There are some near the airport, but they are not obvious from the main road, IIRC.