Learn English – Word used to describe a feeling felt by two people at the same time

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.

Would a "nagging feeling" or "lingering doubt" suit your purpose?