First lets set the reference. The following is my opinion and in no way meant as a knock against you or your preferences. I believe what YOU like is most important. Not what someone else likes. With that out of the way let's roll.
As far as what's relevant to your question, I have a Fulla 2, HD600's, and HD598cs'.
I bought my 598cs' used for $35 so I kept them. That being said I really, really don't like how they sound. To me they are congested, muffled, the mids are recessed, and the bass is mid-bass boomish, but there's no sub-bass slam. I'm saying this so you understand how our opinions differ so you'll know what to make of the rest based on how you think your current cans sound.
The Fulla 2 will power the HD6XX or HD600 fine so you're good there. To me the Fulla 2 is clean and mostly neutral with a hint of brightness but a little thin sounding or cold. The DAC in it is better than the amp but that's nothing to fuss over or sweat. It's a perfectly good DAC/amp. None are "perfect".
Given what you're coming from with the HD598cs' I would steer you to the HD600's. Yes, you will hear a quantifiable difference. I can't say what you will like more but comparing the 598cs & 600, which of course is comparing a closed headphone to a open back headphone, the things you will notice are as follows.
Much more air in the sound, better sub-bass extension, better bass texture, richer vocals, less dark (veil/muffled/subdued) sound, better mids, better highs, less congestion, better instrument separation, MUCH better strings, drums, cymbals and decay, much better detail and many more nuances. I could go on but it is in it's entirety an upgrade vs what you have. Much more of a reference grade sound and more premium build and comfort.
Here's what you might not like. The sound will be more open, you will hear more of your surrounding and need a quieter listening environment to fully enjoy them at times, the heavy mid bass will be gone, they won't be as warm sounding or as dark (though they do have noticeable warmth even though they are very neutral). They won't have the same kind of bass. The bass to me is way better on the HD600's. But, you don't get it if it's not there. The trade off is you get more of it (this needs an example) like, let's say you were listening to a bass drum, the 598cs may give you more bass as an instant "boom", but the HD600 gives you that same boom, but less loud, with more texture, and then you hear that "boom" decay. Hope that kinda makes sense to you. It gives you "more" bass by giving sound to lots of low frequency bass or bass decay or texture you don't normally hear on lesser headphones.
Here's the thing. It's about what you like though, I do recommend them wholeheartedly the HD600 (I don't have the 6XX, I bought the 600's because they are staples in the industry). They offer what you are looking for as far as an upgrade. The HD6XX is a little darker sounding but it does have more bass (boom) at the expense of being less neutral in it's sound. Either would be great and a huge step up for you sound wise. I hope this was helpful.
Lastly, you may want to consider the HD598sr. I know they're still 598's but they are open back and have different drivers. They have more treble than what you currently have, and are more neutral than what you currently have too. No, they aren't as good as the HD600, but since you like what you have, and these keep some of those qualities while offering a good amount of what you're looking for for $139, it may be worth considering as well.
