Just added Sonarr as a docker and am updating tv shows. Noticed I am now locked out of my downloads folder. Do I simply need to run Docker Safe New Permissions or do I have a permission problem directly in the Sonarr docker?

You are not locked out of your downloads folder. You simply don't have access to those three folders inside it. Presumably your new docker is responsible for that. New Permissions would be a temporary fix but your next batch of downloads would have the wrong permissions again. So you need to fix the cause, not the symptom. I don't use downloaders so I can't help there. Maybe search for "Sonarr permissions"?