You have much to learn young Padawan. :jk:
Is a good decision to wait before buying a DAC and Amp, specially amp which has a symbiotic relationship with your headphones. But is inevitable and needed step.
However, I have to say... you should avoid using PC Onboard Audio, specially Realtek and sorry to tell you but you haven't hear HD560S true sound capabilities either like at all -no exaggerating, Onboard audio is such big @#$% performance funnel for your headphones and on top of that you manage volume via software is even worse. That's al right I get it you trying to do your best what we have at hand.
But friend, you need a decent DAC and Amp to start "tasting" and testing headphones until you find the sound firm you prefer.
My humble suggestion, without breaking the bank. Get yourself something like Mayflower Arc Mini which has a HI-Fi DAC and powerful enough transparent amplification to power something difficult yet pretty good like HD600 or HD6XX nicely.
Yes is way better than Fiio offerings at the same price range.
Now if money is not an obstacle but you want to have a nice baseline without investing much: Schiit Modi 3 + Schiit IEMagni 3 or JDS Atom or ifi Zen CAN, are the best entry level to taste any headphone in the market -exception Susvara needs a powerplant-.
Yes, HD6XX are very intimate headphone (meaning vocals sound like you were the mic) and its soundstage is narrow, but no useless for games, how spacious or narrow a hifi headphone must be, depends on the listener. Just like glasses some will find its "closer" midrange(where foot steps are) better than AKG 712Pro or HD800S which could feel way to spacious.
XS is what Sundara aspire to be, similar to Ananda, but with better sound separation, soundstage are about the same. Punchy bass is something that Sundara does better than Ananda, and XS also achieves. High midrange on Ananda sounds like very exciting like throwing much detail in your face... and is the biggest difference on XS, where is more smooth and ease... I prefer that... specially on female vocals, XS sounds more natural than Ananda which can feel nasal sometimes.
Anyways is a big rabbit hole.
but you need an entry DAC and Amp if you are going to dive in.
I know insist way too much but Mayflower ARC MK2 is all you need to for gaming and casual hi-fi music listening.
