If you're not rooted and/or don't want to mess around with the youtube app, sideload TubeMate. When you switch to full screen, it'll give you resolution options. And of course, TubeMate's main function is that it allows you to download the video to your device. Downloading youtube videos doesn't jive with Google's TOS, so you have to do a search and sideload it; it's unavailable from the Play Store. I can't get FHD to work for playback, even on my LG 8.3 tablet ("not supported); it maxes at 720p. However, I can download at FHD and watch it with a 3rd party player (but requires MP3 Video Converter to also be installed).
There's also SuperTube (available from the Play Store). No downloading of course, but it lets you select the resolution up to 720p. The nifty thing with this app is that it's also a floating player, which means you can watch a video on top of another app or your homescreens.