I have come to know about $\mathcal O_K$ with $K = \mathbb Q(\sqrt{69})$. The norm function needs to be adjusted to absolute value, as is the case with other real rings, but it also needs to be adjusted for primes lying over $23$.
Is there another real quadratic integer ring that is Euclidean but not norm-Euclidean, but for which the norm function can, with two adjustments (absolute value and something else), be used as the Euclidean function? And if so, what is it?