I'm not in the audio business but the datasheet lays out the THD at < 0.2% for the sine wave in the audio spectrum. I have no idea whether this is high, medium or low. Page 3 here:
file:///C:/Users/Richa.DESKTOP-1K83UAF/Downloads/DG1022_datasheet.pdf
I would expect all digitally synthesized sine waves to have a higher level of distortion. One solution might be a low pass filter on the output to knock down any harmonics. But harmonics of what? If the tone is 1 kHz, there is room for a lot of harmonics if the filter is set to 20 kHz or so.
eBay is loaded with audio signal generators of dubious quality and no warranty. I didn't see any of the old time HPs and that's where I would go if I was interested in quality sign waves.
Then again, is the function generator just being used to inject a signal and chase it down the rathole? In that case, a small amount of distortion might not matter. If, OTOH, the generator is being used as a source in the design of low distortion amplifiers, it might not be the right tool.
I actually want to use the arbitrary feature to derive signals representing potholes in the roadway that I can feed to my analog computer simulation of an automobile suspension. That would be very cool!