We constructed two new models for deuterium and spin-state chemistry for the purpose of modeling the low-temperature environment prevailing in starless and pre-stellar cores. The fundamental difference between the two models is in the treatment of ion-molecule proton-donation reactions of the form $rm XH^+ + Y longrightarrow X + YH^+$, which are allowed to proceed either via full scrambling or via direct proton hop, i.e., disregarding proton exchange. The choice of the reaction mechanism affects both deuterium and spin-state chemistry, and in this work our main interest is on the effect on deuterated ammonia. We applied the new models to the starless core H-MM1, where several deuterated forms of ammonia have been observed. Our investigation slightly favors the proton hop mechanism over full scrambling because the ammonia D/H ratios are better fit by the former model, although neither model can reproduce the observed $rm NH_2D$ ortho-to-para ratio of 3 (the models predict a value of $sim$2). Extending the proton hop scenario to hydrogen atom abstraction reactions yields a good agreement for the spin-state abundance ratios, but greatly overestimates the deuterium fractions of ammonia. However, one can find a reasonably good agreement with the observations with this model by increasing the cosmic-ray ionization rate over the commonly-adopted value of $sim$$10^{-17},rm s^{-1}$. We also find that the deuterium fractions of several other species, such as $rm H_2CO$, $rm H_2O$, and $rm CH_3$, are sensitive to the adopted proton-donation reaction mechanism. Whether the full scrambling or proton hop mechanism dominates may be dependent on the reacting system, and new laboratory and theoretical studies for various reacting systems are needed to constrain chemical models.
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