If you are planning to use the BA or Qantas lounges, I would suggest you bring a copy of the OneWorld lounge access rules with you. The BA lounge at least has contract staff doing the sign-in, and they tend not to be that familiar with the more unusual cases.
The BAEC FlyerTalk lounge details sticky has a good comparison of the BA and AA lounges (scroll down to DFW). The BA lounge is a lot smaller, but it has free food, free drinks and champagne on request. The AA D lounge has a very small range of free drinks (though BAEC guests get drinks vouchers for more, but I think AA ones don't), and no free food. The AA lounge in D is much larger, has better views over the airport, and is open longer.
I believe that the AA lounge in terminal D is the largest and nicest, but I'm told the ones in other terminals aren't that bad, though suffer from a similar lack of free food / limited free drinks. The BA lounge and AA in D both have showers, though I've not tried either. I'm not sure about the other AA clubs, but I'd guess the D one would get the most use for that.
Assuming you're flying into DFW in the afternoon, I'd suggest you head first to the BA or Qantas lounge. Get some food there, and sample the champagne. Nearer the time of your connecting flight, drift over to that terminal, and have a last drink + use of wifi in the AA lounge near your gate. I would advise against waiting in D until the last minute, then dashing from a D lounge to a different terminal gate, as there can be a wait for trains + long walks and it's not much fun to have to run...
If you don't mind wading through the odd off-topic bit, FlyerTalk often has good comparisons. See posts like this or this or this for recent discussions on them.
Oh, and there's also a Priority Pass lounge in Terminal D, next door to the BA one. It has a few snacks, an average selection of free drinks (which did include a very nice local beer!), dreadful wifi but a few different areas to sit in. Probably not that relevant for you, but might be suitable for others!