OK. $500 gives you quite a few options, but tbh, I don’t think I recommend you drop that much on a stack of electronics right now. Do you have much experience with other headphones? If not, I think you might get more mileage out of a modest upgrade to amp and dac and then throw the bulk of that money at a different set of headphones. With rare exception, changing headphones is going to change your sound the most. Once you have headphones you know you like, then you start chasing the amp, then dac that make them sing the best. In general, I recommend you go headphones (or speakers) > amp > dac it spending priority.
If you know you truly love the 6XX already, the best you thing can do for them is tube them. The 6XX absolutely jumps to life off a tube amp. On the low end of price there is the Little Dot mkii. One step up is the Darkvoice 336SE (which Drop sells for ~$200 about once per quarter). Those are not the best tube amps by any stretch, but they will transform your 6XX in ways you really have to hear to believe. If you go this route, then any DAC from the SMSL M100 or Sanskrit 10th Anniversary Edition, Topping D10, D10s, or E30, or EarMen Donald DAC, or JDS Labs Atom DAC will do the job just fine. Again, if your goal is to explore headphones or learn what the true sound of a headphone is, steer clear of the Zen. Don’t get me wrong, the Zen is a fine product. I have one and I quickly realized its strengths. But IMO you have to know you want its sound to go that route.
Want to try different headphones? Grab any of the DACs I just mentioned, and any of the JDS Labs Atom amp, Schiit Magni 3+ or Magni Heresy, or Monolith Liquid Spark for an amp (my personal recs are either the 3+ or the Spark, but they’re all really excellent for the price). In theory, you can get such an amp and dac combo for $200. Now you have up to $300 to try new headphones. Like a thick, bassy sound? Look at Meze 99 Noir. Like a bright sound with lots of detail and great soundstage and imaging? Beyerdynamic DT880 600 ohm. A more balanced signature? Massdrop + HiFiMan HE-4XX. A V-signature? Fostex T50RPmk3 w/ Shure 1540 pads. You could even look for a used HiFiMan Sundara for a neutral-bright signature and a slight overall uptick in performance from most on this list.
Last point:
TBH, this isn’t necessary. A well designed unbalanced (aka single-ended or SE) setup and a well-designed balanced setup will have no difference in sound quality. The balanced design is a little more resistant to signal interference, but on a desktop setup that is almost never an issue. For the headphones themselves, balanced or single ended makes no difference at all. It’s with the electronics (amps and dacs) where the difference will show up, if there is one. In general, a balanced unit will sound better when its utilized in balanced mode over SE mode because it was designed around being balanced and SE was thrown in as a convenience. But, with again rare exception, this tends to only show up when you’re spending say $500 on a single piece and not for the whole stack.
Apologies if that’s information overload. Don’t hesitate to ask questions.