Hey I noticed Adam playing a white jazzmaster in this photo: http://flickr.com/photos/thefinestkiss/2530250024/
I recognize it as like Ley’s from the Toshack Highway / Bolts of Melody band. One web chat later and it is confirmed.
I know Adam sold one of his jazzmasters a time ago and I do remember him using both, presumably for different tunings?, on tour back in the day so this all fits.
Adam mostly sticks to the jazzmasters but had a silver guitar dedicated to the other Jesus as I remember. Jimmy plays a Les Paul primarily and I think breaks out a telecaster on some songs.