All depends on if you are going for a new-big band sound or the old stuff. If it's the old stuff, the possibilities are numerous. Drums - man,
Gene Krupa baby. Hands down. I can't even begin to list all my other favorites right now, and will probably have to come back to address this later. Sax? You want some really blowin' jump-blues, style swing? That'd be
Louis Jordan. Another awesome sax cat, same style -
Red Prysock (Arthur Prysock). Since I'm in that style, I can't leave out
Tiny Bradshaw or
Wynonie Harris. Those cats were balls-to-the-wall amazing. Can't also forget the good-fella of the style who came along later,
Louis Prima.
That's not really traditional, white-bread big-band. But, it's a damn fine greasy-jump-jive-soul-touching sytle of swing that will get your ass on the floor and get your sugar-mamma flyin'.
Now, if you're going for a kind of new-school approach, there's a bunch of good new cats who really epitomize the old-style, Benny Goodman, Glen Miller, Bix school of thought. Check out any of the following:
Bill Elliot Swing Orchestra (probably the most authentic cats around)
Blues Jumpers (amazing)
Indigo Swing (amazing)
Swing Session (amazing)
Stompy Jones (came out of Swing Session)
George Gee's something another (look up George Gee and that'll do)
Royal Crown Revue (BBVD's twin brother of a different mother)
Brian Setzer Orchestra (as others have mentioned, BUT, start SOLELY with his 1st cd - the self titled one)
Lavay Smith and her Red Hot Skillet Lickers (can't forget the Billy Holiday torch bearer)
Jimmy Fortune (from Canada of all places, but damn good nonetheless)
New Morty Show (Morty was always kind of an idiot, but George Gee above came from these guys...also, you'll find a hellacious version of "Enter Sandman" by Metallica done by NMS)
Steve Lucky (not my cup of tea, but many folks like him)
Big Six (fantastic and leads me to my last point...)
My last point...something else you should definitely check out and a guy who simply doesn't get enough credit:
Bill Haley and the Comets. Damn fine swingin' jump-rock'n'roll. That group the Big Six I mentioned above are the modern-day Bill Haley. Many only know "Rock around the Clock" or "See You Later Alligator," but just go get a very cheap-o greatest hits cd that's out now by BH & The Comets, and you'll get stuff like "Rocket 88" and "13 Women" and you'll see the other cats in the band like Marshall Lytle coming shining through, and see just how amazing they really were, ocassional novelty aside. It'll also have both the sax and drums you're looking for - by the boatload.
That's a good starter. I think. I'm starting to get the buzz of a long-weekend happy hour, so maybe this sounds like crap. But, hopefully it's helped a bit. Good luck.