I started with a Fulla 2 and agree that I really didn't like the pairing with the HD6XX. I upgraded to the Magni 3 (now using the Fulla as a DAC only) and noticed a substantial boost in clarity and bass response. The Magni 3 puts about 11x as much power into 300ohms as the Fulla, so you might be able to get away with the Massdrop O2 (which sits somewhere in the middle) but I haven't tried personally.
Depending on your source, you can potentially get away with just an amp. If you're playing out of a desktop and have a half-decent DAC on your motherboard, and you're not having problems with interference, your most budget solution could be no DAC at all. If you have a shitty motherboard or a laptop, it's probably worth springing for a budget DAC and anything from Schiit or Massdrop is probably fine.