What I've heard is that the Moon is slowly drifting away indeed. At some point, in a "close" future (a few hundreds of thousands of years), full sun eclipses won't be possible anymore because of this. But I've never heard about the Moon getting closer after some point. Now I'm not specialized into this...
The consequence is that the barycenter will drift to a higher point indeed, even if the Moon will have less gravitational influence then.
I haven't watched it all, but this sounds correct to me. But even if the Earth-Moon system becomes a binary system because of its center of mass outside of Earth, that won't drastically change things.
If you want another fun fact, the center of mass of Jupiter and the Sun is above Sun's surface, so following that logic, the Sun and Jupiter should be considered as a binary system too.