So, took a while for them to get here, but to quickly sum up the results…
The amp arrived first, used it for a week or so without DAC on the HD 558s. I think I could notice a difference… but could certainly be placebo. I heard amps won’t help the 558s much anyways.
DAC got then got here, and I’m pretty sure I can tell a difference in noise. Definitely a nice-to-have.
Got the DT 880s, tested them out immediately against the HD 558s (both through amp/dac). The wife and my initial impression was that music sounds better on the 558s, a good test song we used was Panic! At the Disco - Death of a Bachelor. She described it like the sax seemed to lose a lot of it’s… bass-yness I think would be the right. Perhaps the 990s wouldn’t have this issue.
Used it in gaming over the next like… two weeks? Time has passed by strange recently with the quarantine. Holy shit does it make a big difference. Where I was usually like “I hear someone to the left-ish” with a general direction, I’ve now got a pretty good idea as to where they are. Hell, I’ve second guessed myself a handful of times in Apex Legends where I’m like… I thought I heard something over on the left… but my teammate is walking nearby. Maybe it was just that. NOPE!
As a final decision on the music aspect… they are kind-of just different. The 880s definitely seem to have a wider sound stage. Sounds seem to be a bit more clear. That said, I think the 558s (to my uneducated ears) might be better for music. Not enough to swap back though.
TL;DR; Sennheiser HD 558 to Beyerdynamic DT 880
For competitive gaming: Holy shit, so much better. Super worth it.
For music: Mostly just different, probably better upgrade choices for this.