Would "erase all content and settings" disassociate my phone with the carrier?
No. Cellular plans do not get erased since that registration belongs to the SIM card and cellular systems, not something that’s a setting on the phone that can be erased by the OS or a restore.
If you go to

Settings -&gt; General -&gt; Reset -&gt; Erase All Content and Settings

This will restore the device to an 'Out of the Box' state and allow you to start the process of set up again. It will not deactivate the device as far as the carrier is concerned and you can still make and receive calls and SMS but will lose contacts and call history, etc....