Why does CiviCRM keep logging me out of the website admin?
Check that the URL configured for your WordPress site matches the URL in your CiviCRM civicrm.settings.php

If these have different URLs, you will appear to be logged out when you move "between sites". This sounds like it could match what you're describing.

The URLs must match for all of protocol (http vs HTTPS), domain (example.org vs example.com) and hostname (www.example.com vs example.com).