The dynamics of the organic components of the deep eutectic solvent (DES) glyceline are analyzed using an array of complementary nuclear magnetic resonance (NMR) methods. Fast-field cycling 1H relaxometry, pulsed field gradient diffusion, nuclear overhauser effect spectroscopy (NOESY), 13C NMR relaxation, and pressure-dependent NMR experiments are deployed to sample a range of frequencies and modes of motion of the glycerol and choline components of the DES. Generally, translational and rotational diffusion of glycerol are more rapid than those of choline while short-range rotational motions observed from 13C relaxation indicate slow local motion of glycerol at low choline chloride (ChCl) content. The rates of glycerol and choline local motions become more similar at higher ChCl. This result taken together with pressure-dependent NMR studies show that the addition of ChCl makes it easier to disrupt glycerol packing. Finally, a relatively slow hydroxyl H-exchange process between glycerol and choline protons is deduced from the data. Consistent with this, NOESY results indicate relatively little direct H-bonding between glycerol and choline. These results suggest that the glycerol H-bonding network is disrupted as choline is added, but primarily in regions where there is intimate mixing of the two components. Thus, the local dynamics of most of the glycerol resembles that of pure glycerol until substantial choline chloride is present.