With your budget, I would just focus on a good amp and forego the DAC. Unless you have a very poor source, you won't notice much of a difference until you start spending a bit more.
With that said, I recommend the Fiio A5. It's very powerful and portable. Compared to the Fulla 2, it has almost 4x times the power at 300 ohms (150mW) and has more voltage output (14.96Vpp = 5.2Vrms). The Fulla will be maxed comparatively whereas the Fiio will have plenty of headroom and you won't have to worry about your music clipping.
If your budget was a bit more ($200), I'd recommend the iFi nano Black Label. It's also powerful enough to push the 6XX (output of 3.5V) and has a better DAC section and supports more filetypes and bitrates over the Fulla. It's also portable and battery-powered (about 10 hours) and has some other features built-in: a USB purifier (helps eliminate jitter) and iEMatch for sensitive IEMs that tend to hiss. It's also balanced with 3.5mm TRRS if one was inclined. The nano also works well in desktop mode since you can fully disable the battery and run it right off USB (reduces battery wear).
If you're on an absolute budget, the Fulla is good for its price. However, if you spend just a bit more, you'll get more on your investment with better quality sound.