Modernize Phame process handlers

Summary: Converts Phame to use handleRequest where appropriate.

Test Plan: Write some blog posts, publish, edit, view.

Reviewers: btrahan, epriestley

Reviewed By: epriestley

Subscribers: epriestley, Korvin

Maniphest Tasks: T8628

Differential Revision: https://secure.phabricator.com/D13677