How do I switch to a completely new Apple ID?
When I changed my ID on my iphone, I didn't see an "edit" option either.  So maybe my experience will help you, I hope.  After going to Settings/iCloud, I was told to scroll down to the bottom of the page where it says DELETE ACCOUNT.  After deleting my son's account from my phone, I was able to enter my own ID.  It took a few tries and some time while it tried to validate the new ID and then it told me to go to my email on my computer to respond for verification.  Good Luck!