I've experienced what you've described. I have to do a lot of public speaking and critical thinking at my job, and sleep depravation hinders me.
When I was in H.S., I was prescribed sleeping pills - - I slept more, but I was even more lethargic, so I stopped.
In college, I discovered meditation, and that worked pretty well for me.
Also, I'm the type of person that can't sleep to TV. If I put in a movie, no matter how tired I am, I stay awake. TVs are like crack sometimes.
It sounds like TV affects you similarly - maybe try cutting that first. Reading helps me fall asleep, but I've known people whom it keeps awake.
If you do have to speak and think while sleep deprived, keep in mind that things will move a little slower. Accept it, and just take your time. It's usually when I try to go at "normal" speed that I develop a stutter.