No, it is not necessary. The web adaptor allows you to integrate ArcGIS for Server with your web server. You can still use any other published services directly from your ArcGIS server. For example:
is the internal path to our collection of rest services
is the path to our rest services from our web adaptor
All services function exactly in the same way.
Thank you for the quick updating. I am wondering if your print url in your web mapping application is linked from internal server service or external rest service (web adaptor). Or the print tool is working from both server services.