Do I own the full rights to my purchased Logo?

Absolutely, securing your logo means acquiring exclusive ownership over the creative assets. 
You receive a boundless, international, and non-royalty permit (as per our Terms of service).

Even with a termination of your membership, you retain the benefits of our service until the completion of your payment period.

Upon the termination of your package, you continue to hold total ownership for your previously acquired logo designs and related resources. Simply put, you remain free to deploy your previously secured logo files on your website, social networks, and any manner that you prefer.

Point to note: A "TM" or trademark isn’t included in the Zenn subscription.
However, as pointed out beforehand, securing a membership means acquiring total ownership over the design regardless of the continuity or discontinuity of your membership.