I worked out most of this by watching John Oates play an acoustic version. It may not be 
precise to the studio recording, but it's darn close.

Here are the shapes John uses:

EADGBE

A: x02220
E: 476454
D: 557775
D6: 557777
Bm: 799777 (sometimes he plays 797779)
Bm7: x24232
C#m7: x46454
F#m: 244222

... Though, for most of those chords, the standard shapes sound fine, too. Also, the D6 can be substituted with a standard D and still sound good. (Personally, I like substituting a Dmaj7, played like this: xx0222.)

Intro:

A E D Bm C#m7 D

A (x02220@1)   E (022100@1)           D (xx0232@1)  BmOpening (x24432@1)night,  nothing new Atlanta
         A (x02220@1)                 EInto (022100@1)the spotlight, one more time
                D (xx0232@1)  BmJust (x24432@1)in time to play
   A (x02220@1)            ETo (022100@1)one man at an empty table
       D (xx0232@1)      C#m7 (x4242x@1)    Bm7He (x20202@1)was drinking down the pain
             Bm7 (x20202@1)   C#m7 (x4242x@1)      D6All (xx04xx@1)he could say was his lady's name

A (x02220@1)   E (022100@1)              D (xx0232@1)        BmOh (x24432@1)Camellia, won't you take me away

A (x02220@1)   E (022100@1)              D (xx0232@1)        BmOh (x24432@1)Camellia, won't you take me away

A (x02220@1)   E (022100@1)               D (xx0232@1) BmAfter (x24432@1)the show, when the room was empty
No, he wouldn't go
So I asked him why he called her name
It seems she was some magic one night
     D (xx0232@1)       C#m7 (x4242x@1)  Bm7With (x20202@1)something for his pain
            Bm7 (x20202@1)        C#m7 (x4242x@1) D6But (xx04xx@1)all she left was her pretty name...

A (x02220@1)   E (022100@1)              DOh (xx0232@1)Camellia, won't you take me away
   Bm (x24432@1)              C#m7To (x4242x@1)paradise tropical moon
           F#m (244222@1)                   Bm7 (x20202@1)   DDon't (xx0232@1)you leave me sitting here in Atlanta

Repeat chorus until the end!