No the interior ministry kept the non-Nazi part of their emblem. They also kept the name, which also isn't Nazi but the name of the Azov Sea (east of Crimea, north of the Kerch straight). When Azov got integrated into state structure tons of Nazis left because they didn't want to be under state control and be told what they could and couldn't do politically etc, some stayed but the state imposed strict "don't do Nazi things" controls. In the beginning (2016 or such) it was about 20% Nazis left over so even back then a minority, there's been churn and growth since then so it should be quite a bit lower by now.

The old pre-national guard Azov was definitely a Nazi cesspool, yes, and alas one of the very few capable fighting forces at the start of the Russian invasion (for the record, that's 2014). Then regular people joined because they also wanted to fight and could look past the iffiness, that's why Azov already hadn't been purely Nazi at the time the Ukrainian state integrated and denazified them.

But all that nuance is lost in both Russian propaganda and also western media, where some libs found themselves a juicy topic to sell clicks with.