The RME ADI seems like a pretty technical piece of equipment ? What did you adjust for better deeper bass for gaming ?
I just swapped to the HD800S this year, and while it sounds good, I don't think it made me a better player than when I had the 6XX on.
As far amp and gaming, you want something accurate, which means something that measures well. The subjectivists on other forums call this cold and analytical.
I use a THX AAA 789, but you don't even need to spend that much. Just get the cheapest amp that measures well and go line in from the motherboard line out. If you insist on a DAC or a DAC/Amp combo, just get the cheapest that measures well, if this is mainly for gaming.
I use Equalizer APO on windows for any EQ that I need; I generally use Harman target + 2 dB low shelf at 105 Hz for a bit more bass. I turn off the EQ for gaming though, it's just my music preference.