High-resolution James Webb Space Telescope (JWST) observations can test confusion-limited Hubble Space Telescope (HST) observations for a photometric bias that could affect extragalactic Cepheids and the determination of the Hubble constant. We present JWST NIRCAM observations in two epochs and three filters of >320 Cepheids in NGC 4258 (which has a 1.5% maser-based geometric distance) and in NGC 5584 (host of SN Ia 2007af), near the median distance of the SH0ES HST SN Ia host sample and with the best leverage among them to detect such a bias. JWST provides far superior source separation from line-of-sight companions than HST in the near-infrared to largely negate confusion or crowding noise at these wavelengths, where extinction is minimal. The result is a remarkable >2.5x reduction in the dispersion of the Cepheid period-luminosity relations, from 0.45 to 0.17 mag, improving individual Cepheid precision from 20% to 7%. Two-epoch photometry confirmed identifications, tested JWST photometric stability, and constrained Cepheid phases. The P-L relation intercepts are in very good agreement, with differences (JWST-HST) of 0.00 +/- 0.03 and 0.02 +/- 0.03 mag for NGC 4258 and NGC 5584, respectively. The difference in the determination of H0 between HST and JWST from these intercepts is 0.02 +/- 0.04 mag, insensitive to JWST zero-points or count rate nonlinearity thanks to error cancellation between rungs. We explore a broad range of analysis variants (including passband combinations, phase corrections, measured detector offsets, and crowding levels) indicating robust baseline results. These observations provide the strongest evidence yet that systematic errors in HST Cepheid photometry do not play a significant role in the present Hubble Tension. Upcoming JWST observations of >12 SN Ia hosts should further refine the local measurement of the Hubble constant.