Why is my Google storage full when my Google Drive doesn't look that big?
Short answer: Google storage is shared between Gmail, Google Photos and Google Drive. Storage Studio can move old Drive files off to your drive; Google Photos isn't available in the app right now, so use Google Takeout for photos.
Because your Google account's storage is shared: Gmail, Google Photos and Google Drive all count toward the same total. A small-looking Drive can still leave the account full if your mail or photos are large. Google's own storage page shows the split.
Where Storage Studio can help is the Drive part: connect Google Drive and move old files to your external drive or another account with Data Transfers. Choose copy first if you want to check the result before anything leaves Drive.
Google Photos can't be added to Storage Studio right now, so for photos the route is Google Takeout — see getting photos out of Google Photos.