The unprecedented rest-frame UV and optical coverage provided by James Webb Space Telescope (JWST) enables simultaneous constraints on the electron density ($n_{\rm e}$) and temperature ($T_{\rm e}$) of ionized gas in galaxies at $z\,\gt\,5$. We present a self-consistent direct method based on multiple O III] ($\lambda \lambda$1661, 1666) and [O III] ($\lambda \lambda$4363, 5007) transitions to characterize the physical conditions of the high-ionization zone. This new approach is insensitive to a wide range of $n_{\rm e}$ due to the high critical densities of the O III] and [O III] transitions. Applying this technique to six galaxies at $z \sim 5$─9, we find electron densities up to $n_{\rm e}$ $\sim 3 \times 10^{5}$ cm$^{-3}$ and temperatures of $T_{\rm e}$ $\sim 20\,000$ K in systems at $z\,\gt\,6$. Accounting for these self-consistent densities changes the derived $T_{\rm e}$ and modifies the inferred metallicities by up to 0.29 dex relative to previous estimates. We discuss the reported N/O overabundances in our high-z sample, driven by the high N$^{3+}$/H$^{+}$ ratios inferred from the N IV] lines. We point out that a $T_{\rm e}$-stratification, in which the N$^{3+}$ zone has a slightly higher $T_{\rm e}$ than $T_{\rm e}$([O III]), could substantially reduce the inferred N/O. Quantitatively, if $T_{\rm e}$(N$^{3+}$) were 10 per cent higher than $T_{\rm e}$([O III]), this could induce a systematic overestimation of N$^{3+}$/O$^{2+}$ of nearly 50 per cent. Classical N/O diagnostics such as N$^{+}$/O$^{+}$, due to their critical densities, can significantly impact the inferred N/O abundance in the presence of high-density gas, whereas N$^{2+}$/O$^{2+}$ place these galaxies closer to $z\sim 0$ systems in the N/O─O/H plane. While promising, this method accounts for density inhomogeneities but not $T_{\rm e}$ fluctuations, and relies on UV atomic data in the high-$n_{\rm e}$ regime. Future JWST programs will be key to testing its robustness.