No. You retain ownership. You are only telling the rest of the worldwhere to find your domain on the Internet. It is like giving out your new street address after you move to a new home.
To change your nameservers to point to Avangard Hosting, please change them to the nameservers assigned to you upon signup. If you do not have this information, please contact our support department. After changing your DNS settings, they must be spread to all the DNS servers worldwide. This is known as DNS propogation. Propogation can take up to 72 hours to complete. There is nothing that can be done to speed this up as the main DNS servers around the world update according to their own schedules. During this time your site may appear to be up at times and down at others. It may also be visible for some and not visible for others. This will effect all of your web services.