We use models of stellar angular momentum evolution to determine ages for ∼500 stars in the APOGEE-Kepler Cool Dwarfs sample. We focus on lower-main-sequence stars, where other age-dating tools become ineffective. Our age distributions are compared to those derived from asteroseismic and giant samples and solar analogs. We are able to recover gyrochronological ages for old, lower-main-sequence stars, a remarkable improvement over prior work in hotter stars. Under our model assumptions, our ages have a median relative uncertainty of 14%, comparable to the age precision inferred for more massive stars using traditional methods. We investigate trends of Galactic α-enhancement with age, finding evidence of a detection threshold between the age of the oldest α-poor stars and that of the bulk α-rich population. We argue that gyrochronology is an effective tool reaching ages of 10─12 Gyr in K and early M dwarfs. Finally, we present the first effort to quantify the impact of detailed abundance patterns on rotational evolution. We estimate a ∼15% bias in age for cool, α-enhanced (+0.4 dex) stars when standard solar-abundance-pattern rotational models are used for age inference, rather than models that appropriately account for α-enrichment.