Just to make sure we are looking at the same place, here are the steps to change contact information (Contact name, email, phone, mobile phone, address) in FamilySearch. This is also the place where you select whether you want your information public (each individual item can be public or private):
(Note: I'm using a desktop computer running Chrome browser - this should be browser independent, but my wife has a laptop with a touch screen and instead of her name there are 3 lines in the corner she must click [she is also running Chrome])
1. Log in to FamilySearch
2. Click on your name in the upper right corner
3. Select "Settings" in the drop-down box - you will be taken to your settings page with your name in a banner across the top of the page
4. Just below the banner, click on "CONTACT" (the choices are account, contact, notifications, preferences, connected accounts)
5. Change the information in the respective boxes. When you make a change, a blue "Save" button will become active at the bottom.
6. Click "Save" to save your contact changes. If you con't click the Save button when finished, your changes are not saved.
6. When you click "Save" a green bar will appear near the top telling you your changes were saved successfully.
Hope this helps!