Of course, the Moon (presently) rotates in the same direction as it revolves, and it is gravitationally locked into a 1:1 synchronization because the "tides" in the
formerly molten rock have cooled and solidified, and the resulting elipsoidal shape is permamently oriented with its major axis pointing toward the earth.
I went ahead and helped you too.
Indeed, the
Moon's core is not solid, and that is why it is tidally-locked. Good for you. However, your original statement:
The swinging weight of a molten moon core would either have sent the moon flinging off into space or would have sent it crashing to earth.
Is still not accurate. Several Jovian and Saturnian moons have molten cores (hence, vulcanism) and they have neither been sent "flinging off into space" or crashing into their primaries.