Hi There Wyo. Having led bands, big and small over the years I think one bit of info that might help is this: You've got it right already!
Generally the guys have had tons more experience than you because they've had tons more spare time. (I might get an argument here, but I feel I'm on solid ground) Here's a few things that might make the process easier, and I trust you when you say you can "deliver the goods".
1. And this is crucial!! Get at least 3 or 4 songs written out on charts - in YOUR key, with proper chords and a melody line. Have the appropriate number of copies needed and hand them out. You will soon find which members of the band REALLY know what they're doin', and which ones suddenly are worrying about keeping up with you! I can't emphasise this enough. It will show the band that you know the REAL rules that professionals expect and will put you on an even footing without any aggressiveness that might hurt feelings. Any local pro can do this for you and it will probably cost about 50 bucks.
2. Keep a Sony Pro Walkman with you at all times. Don't go for a cheaper model, and don't go for the DAT one.(you want to be able to play things on anybody's stereo)
Ask the band leader to play their repertoire so you can tape it, and tell him that's because you want to learn the songs RIGHT. He'll be impressed. Then learn them on your time - not theirs.
3. Buy your own mike (with low impedence cord, and hi impedence adaptor). Sure SM57 is still preferred my most vocalists.

Well, there you go. It involves a little investment, but I absolutely guarantee you that if you take care of those things and don't scrimp, and if you're timing and pitch are spot on - the word will get around so fast you won't believe how well you'll be treated. You can also counter the "chick singer" jokes with "drummer" and "bassplayer" ones.
Although I'm a dyed in the wool folkie, I love old time swing, and I wish you luck.
rick