No, because the Moon is in free fall around the Earth. If the Moon wasn't there, you'd continue in the same orbit as the Moon. So the Earth's gravity has no effect on you.
It's a bit like, a droplet of water in the ISS holds together easily. Although it's got the entire pull of the Earth to one side - but because the whole system is in free fall around the Earth, then it makes no difference.
All you feel is the gravitational pull of the Moon.