Deuterium-to-hydrogen (D/H) enrichments in molecular species provide clues about their original formation environment. The organic materials in primitive solar system bodies have generally higher D/H ratios and show greater D/H variation when compared to D/H in solar system water. We propose this difference arises at least in part due to 1) the availability of additional chemical fractionation pathways for organics beyond that for water, and 2) the higher volatility of key carbon reservoirs compared to oxygen. We test this hypothesis using detailed disk models, including a sophisticated, new disk ionization treatment with a low cosmic ray ionization rate, and find that disk chemistry leads to higher deuterium enrichment in organics compared to water, helped especially by fractionation via the precursors CH$_2$D$^+$/CH$_3^+$. We also find that the D/H ratio in individual species varies significantly depending on their particular formation pathways. For example, from $sim20-40$ AU, CH$_4$ can reach $rm{D/Hsim2times10^{-3}}$, while D/H in CH$_3$OH remains locally unaltered. Finally, while the global organic D/H in our models can reproduce intermediately elevated D/H in the bulk hydrocarbon reservoir, our models are unable to reproduce the most deuterium-enriched organic materials in the solar system, and thus our model requires some inheritance from the cold interstellar medium from which the Sun formed.