"WE WERE GOING TO" is a somewhat casual statement of intent to do something.
"WE WERE ABOUT TO" is a more time specific statement of IMMEDIATE intent to do something in the very near (immediate) relative future.
In this example, the meaning could change completely:
"WE WERE GOING TO READ/SIGN/SHRED the contract when the detective phoned us" could mean that you were waiting for his call before you were going to act.
"WE WERE ABOUT TO READ/SIGN/SHRED the contract when the detective phoned us" means that you were very near doing it, when something else happened (he called).