Would it be correct to say that IF you are a member of an organization with full access then you’d have full access to other teams within the same org (search, for example) ? Provided of course items aren’t private?
Not necessarily; you won’t have access to a team unless its privacy level is set to “Public to organization”.
The whole area of permissions and privacy can get a bit complex and is governed by a variety of different settings. Here’s a good post to read on the subject; and I especially encourage reading the pages in the Asana documentation that are linked-to within the post:
Also, while you didn’t mention guest users, if you will have any guests, you might be interested in this post which discusses what a guest can and can’t access in different scenarios: