... and that means I am doing it right, correct?

Apparently, I was under the (wrong) impression (up until recently) that as long as I did exercise in intervals of higher and lower intensity... that THAT was HIIT. Then I did some reading and found out that the high intensity intervals have to be pretty much the hardest your body can work, and that you shouldn't be able to last much longer than 30 seconds at that intensity.

So I have started sprinting as fast as I can during those intervals. When I get to the end of them, my legs feel like jelly, my heart is beating like crazy, and I feel nauseous. All of this goes away within the next 30 seconds, however....

So, that's the right way to do it, correct?

Yes True high intensity intervals do not last longer than 15 (maybe 20) minutes in total. Anything more than that is just "intervals" and not HIIT.

Enjoy it, but remember no more than 2 (maybe 3) times per week maximum or else you will begin to feel pain/injury in your feet.

... note to self ...

It's the same thing with tabatas. The last minute should feel like absolute ****. The original study was only 4 minutes long, so I always get suspicious about tabata circuits.