As far as my experience more a good DAC changes the sound as much as a good amp. Different DAC chips have different sound signatures.
You'll often hear Wolfson DACs sound more natural. Etc.
Onboard DACs are really cheap and there's a lot of electrical noise. I've always found even a cheaper Fiio external DAC to be superior and in some cases help more than a lower end amp.
The biggest change in sound will come from the headphones though.
I'm not really sold on the DAC making different sound on level with amps. Amps can enforce actual changes to frequency response depending on the output impedance and the impedance curve of the headphones. Innerfidelity is currently (has?) conducted DAC blind testing and I don't think the results are conclusive enough to go one way or the other.
As for onboard DACs being cheap, I've not found any EMI from any modern, decently priced mobo that I've heard. Yes, if you get EMI, for sure, get an external DAC, but if you don't, I personally cannot see there being a large enough difference to justify the price.