Google Sites transfers from one domain to another will require manual intervention by the owner of the site.
If you know of any Google Sites, please make note of them and where the Google Site file is currently housed.
NOTE that transferring a Google Site from eng.ucsd.edu to ucsd.edu domain will change the published URL.
IF you need to keep the currently published URL and you do not have a custom URL that you have assigned to the site, contact OEC for instruction on how to keep the current URL in place.
IF you use a custom URL for accessing the site, that OEC has helped configure for you, then it is likely that the Google Site can be moved to the ucsd.edu Google domain and republished - custom URL would point to new location. Please contact OEC for assistance.
If the URL does not matter:
1. Unpublish the Google Site from within your @eng.ucsd.edu account
2. Once you've made your MyDrive or Shared Drive move, then you replublish the Site while signed in as your @ucsd.edu Google account
IMPORTANT: If you are still using a Google Classic Site, note that this style of Google Site is going away - here is more information: https://support.google.com/sites/answer/7035197