Book-Identification: Axe wielding protagonist and a dragon

I know this is an old question, but I just saw it. Maybe this will still help...

Perhaps Dragonsbane by Barbara Hambly (1985)?

There is a kingdom and the northern areas are more or less cut off and decaying. A somewhat foppish teenager (Gareth) who turns out to be royal, comes to ask the local lord (John Aversin), who had previously killed a dragon, to help because there is a dangerous dragon I think near the capital.

John and Jenny Waynest, who has magical training and abilities, agree to go.

The plot is complicated; the Wikipedia article summarizes it, but they do fight the dragon and John is armed with an axe.

He is not fighting to protect his city though, but someone else's, so I am not sure it is a match. Also, you mention that there was a dragon in the story, but not that the axe fight was against the dragon. But there is a dragon and a pages long fight scene where the protagonist wields an axe.

Things that might help you recognize it:
The north lands are a very rough place. Where Gareth comes from, and they later go to, is seemingly much more refined (Gareth is shocked to find his hero to be a seeming country bumpkin). But the "cultured" culture is actually full of petty spite and all sorts of nastiness.

There is another race (dwarves? elves? I don't remember) whose magic comes from something called "the Deep", and one of the more nasty characters is trying to get control of it (Zyerne, I think)

Two more hints, but they are spoilers: