I know that when a load draws too much power the voltage of the grid at the fundamental frequency begins to sag. Though my understanding is that loads can't use the power generated by harmonics, so does that mean the grid voltage will only sag at the fundamental frequency and the amplitude of the harmonics won't?
No, it' doesn't. It depends on the characteristics of the load. If resitive, for example, then it presents the same impedance to all harmonics. Since the impedance of the electric transmission system generally goes up with frequency, a resistive load will actually reduce the harmonics more than the fundamental.