I ran into this as well and it was annoying the heck out of me.
I noticed that even from the stock browser, youtube vids want to launch in the youtube app, even if you tell it to use the browser to perform the action. You can try copying and pasting or forcing the yt link into the browser address bar, it still wants to run the yt app for some reason.
So what I did was go to the youtube full site directly, then located the video I wanted, and then I can select the resolution and go fullscreen.
HUGE difference and the HD stuff looks amazing. You'll know you're viewing in full HD glory as the play bar and menu options at the bottom of the screen will be super small.
Still haven't figured out a way to get it just go to full HD in a single click, but I need to get back in there and monkey with the settings.