We present improved cosmological constraints from a re-analysis of the Dark Energy Survey (DES) 5-year sample of Type Ia supernovae (DES-SN5YR). This re-analysis includes an improved photometric cross-calibration, recent white dwarf observations to cross-calibrate between DES and low-redshift surveys, retraining the SALT3 light-curve model and fixing a numerical approximation in the host-galaxy colour law. Our fully recalibrated sample, which we call DES-Dovekie, comprises $\sim$1600 likely Type Ia SNe from DES and $\sim$200 low-redshift SNe from other surveys. With DES-Dovekie, we obtain $\Omega _{\rm m} = 0.330 \pm 0.015$ in flat Lambda-cold dark matter ($\Lambda$CDM) which changes $\Omega _{\rm m}$ by $-0.022$ compared to DES-SN5YR. Combining DES-Dovekie with cosmic microwave background data from Planck, Atacama Cosmology Telescope, and South Pole Telescope and the DESI DR2 measurements in a flat $w_0 w_a$CDM cosmology, we find $w_0 = -0.803 \pm 0.054$ and $w_a = -0.72 \pm 0.21$. Our results hold a significance of $3.2\sigma$, reduced from $4.2\sigma$ for DES-SN5YR, to reject the null hypothesis that the data are compatible with the cosmological constant. This significance is equivalent to a Bayesian model preference odds of approximately 5:1 in favour of the flat $w_0 w_a$CDM model. Using generally accepted thresholds for model preference, our updated data exhibits only a weak preference for evolving dark energy.