We have used a new discharge-flow apparatus with modulated-beam mass spectrometric detection to study product-branching fractions for the reactions of F atom with methanol and deuterated methanols. Despite the fact that formation of hydroxymethyl radical is favored both energetically and on the basis of hydrogen atom statistics, we find enhanced formation of the methoxy radical in this system. Specifically, the fractional yields of methoxy radicals at room temperature for the two isotopically mixed methanols are X(methoxy) = 0.56 +/- 0.03 for the reaction F + CD3OH --> products and X(methoxy) = 0.37 +/- 0.02 for the reaction F + CH3OD --> products, where the error bars are 2-sigma estimates of statistical errors. Measurements of the fractional yields of methoxy radicals for these two reactions at 830 and 960 K are consistent with a temperature-independent product-branching fraction. Relative-rate measurements for HF and DF formation from the reactions F + CH3OH,CD3OH --> products allow us to calculate site-specific relative reaction rate coefficients and kinetic isotope effects, as well as room temperature methoxy radical yields of X(methoxy) = 0.6 +/- 0.2 (2-sigma) for the reaction F + CH3OH --> products and X(methoxy) = 0.4 +/- 0.3 (2-sigma) for the reaction F + CD3OD --> products. Reasons for the anomalously high methoxy radical yields are discussed.