What other units, besides the 82nd, 173rd, and the 101st, take airborne recruits?

The 101st isn't Airborne at all.
82nd, 173rd in Italy and Germany, and 4/25 in Alaska are the main ones if you're a combat MOS.
If you're a support MOS, have a GT of 100 and a Secret Clearance you could also go to one of several different units at Bragg, SF Groups at JBLM, Carson, Eglin, Campbell Okinawa or Germany. Depending on your MOS there are a dozen other places you could go.
If you try out and get selected you could go to the 75th Ranger Regiment in Benning, Savanah, or JBLM. Or you could try out for the 160th Night Stalkers and be in Savanah, JBLM or Campbell.
Depending on your MOS there are many places. Most likely though, you're going to the 82nd first. You can inquire all you want, but they're going to send you wherever they need you.

Airborne in your contract means nothing to a particular unit. Just means you'll get your wings (granted you pass). If you want to get some jumps in - you'll have to find an airborne unit. Also - just because a unit says "airborne" - as SFC Jason Boyd pointed out - doesn't mean they are (A). 101st - sorry. You'll be air assault (fun school - but have never used any of the skills thereafter).