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The²³Na(α,p)²⁶Mg reaction has been identified as having a significant impact on the nucleosynthesis of several nuclei between Ne and Ti in Type Ia supernovae, and of²³Na and²⁶Al in massive stars. The reaction has been subjected to renewed experimental interest recently, motivated by high uncertainties in early experimental data and in the statistical Hauser-Feshbach models used in reaction rate compilations. Early experiments were affected by target deterioration issues and unquantifiable uncertainties. Three new independent measurements instead are utilizing inverse kinematics and Rutherford scattering monitoring to resolve this. In this work we present directly measured angular distributions of the emitted protons to eliminate a discrepancy in the assumptions made in the recent reaction rate measurements, which results in cross sections differing by a factor of 3. We derive a new combined experimental reaction rate for the²³Na(α,p)²⁶Mg reaction with a total uncertainty of 30% at relevant temperatures. Using our new²³Na(α,p)²⁶Mg rate, the²⁶Al and²³Na production uncertainty is reduced to within 8%. In comparison, using the factor of 10 uncertainty previously recommended by the rate compilation STARLIB,²⁶Al and²³Na production was changing by more than a factor of 2. In Type Ia supernova conditions, the impact on production of²³Na is constrained to within 15%.