They're both known issues with no timeline for a fix.
Dungeon emblems have been bugged since at least deep and they recently also broke the Warlords Ruin emblem.
Stat trackers not updating for things like Light subclasses and Gambit etc have been bugged since Beyond Light launched.
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Although known, if it matters to you as much as me, I feel like mentioning it and trying to get voices heard about it how much it matters to the players is important if its ever going to be fixed, at some point if everyone just stops mentioning it, I fear it will never get fixed...