[killjoywashere]

The Moon orbits the Sun, and is only mildly perturbed by the Earth. Gas drag would decay the Moon's orbit, but it's not at all clear to me it would crash into the Earth. It seems far more likely it would simply fall into the Sun.

[craftyguy]

The Earth and Moon system orbit the Sun, but the Moon is very much in orbit around the Earth.

A more likely scenario is that the Earth's orbit will move outwards as the Sun expands, since stars in the red giant phase(s) lose a lot of mass. It's plausible that the Earth/Moon will not be destroyed, physically. But in that case all life would have been obliterated due to the immense temperatures associated with having the surface of the Sun within a few million miles (or less) of Earth.