I don't hear any obvious distortion in the vocals either.... There's a certain harshness there though.
What first struck me is that the vocals and piano are fighting over the same space, so eqing the piano to reduce it a bit innthe frequencies it shares with the vocals might be worth trying.
Secondly the vocals sound like they've been boosted around the high-mid "telephone" frequencies which often makes voices sound thin and harsh. Try letting more of the lower vocal fundamentals and harmonics through.
Finally, are you sure what you're finding unacceptable isn't simple how your voice sounds? That may sound harsh, but as a pretty ropey singer myself it's something I've had to learn to live with. When we speak or sing what we hear is mostly our voice transmitted through the bones of the jaw and skull. That gives us pretty accurate pitch information but no real idea of tone quality at all. Which is why most people find recordings of their own voice a bit of a shock and a little unsettling when they first hear them.
And don't forget the possibility you may be having a "bad tone day", when things just sound wrong and horrible even if they sounded fine all last week....