Learn English – word for tremendous happiness coupled with relief

There are some good answers here, but nothing quite on the nose, so I'll make my own humble contribution:

Does it at least have a happy ending?

Nope...it's pretty much a downer ending.

This is obviously much less formal than these other answers, but it's what I would use in normal American English speech.

The sensation you describe occurs when you're in free fall, as described on HowStuffWorks:

But in the "free-fall" state, there is hardly any net force acting on you. In this case, the various pieces of your body are not pushing on each other as much. They are all, essentially, weightless, each falling individually inside your body. This is what gives you that unique sinking feeling in your stomach.

Unfortunately, there doesn't seem to be a specific term for it. (Unless there's a really technical medical term that involves your vestibular system somehow.)

On the other hand, that falling sensation (and accompanying twitch) you sometimes get when you're falling asleep? That's a hypnagogic myoclonic twitch.