@billsimon Perfect, it worked! Thanks so much!
It turned out there were probably two ways to do this. The first was to go into this setting you gave me, under each extension, and change the “Send RPID” to NO.
But searching for “Send RPID” led me to this thread, which, coincidentally, had the EXACT same phones (Polycom IP330) that we have, showing CID:EXT on outbound calls:
… which led me to (in case anyone else comes across this): Fixed this by going to Advanced Settings in FreePBX and then under Dialplan and Operational (about half way down the page) I set “Display CallerID on Calling Phone” to False. Now the original caller ID is working on transferred calls and the proper outgoing numbers are appearing on outgoing calls 
Plus it was a single global setting instead of having to modify each extension in case there were any adverse effects. But it was definitely this RPID setting that led me to the fix. Thanks again!