The earth's rotation rate is slowing down because of friction against the tidal bulges caused by the gravitation of the moon (major factor) and the sun (lesser factor). The earth's rotational energy is dissipated as heat, but where is the angular momentum going, and what physical mechanism brings that momentum there?
I suppose it's possible that the moon receives a push into higher orbit, but I can't believe that the net angular momentum of the two bodies is conserved. Most of it is dissipated by the same entropy of the motion of the fluids on earth (ie., the weather, ultimately, heat--as you said above). This takes up the angular momentum the same way the it takes up the energy.
What if the earth had no moon? Are you saying it wouldn't slow down?
And, what happens to the angular momentum when you spin a raw egg? Same thing, only much faster in the case of the egg.