Here is a partial answer that works for my situation, when the full screen mode is for presenting Google Slides (or the rare times I use that Microsoft product that rhymes with "Shower Joint").
What works for me is when going to presentation mode, to use the option for Presenter View. In Google I then close the window with speaker notes, and my slides are in display mode, but it's full in the tab, not window.
For display purposes (projector or say screen sharing from video conference app) it shows like full screen, no window or browser elements visible.
Now I can use Command+`to switch between my slides and another browser window where I am in say my video conference software. I like using this so I can jump from slides back to see the video conference chat.