I think a lot of people get stuck on the 'undead' part and miss the 'dragonborn' element, which is where the real conflict simmers. The basic stuff is obvious—they're a walking contradiction, life and death fused into one being, so loneliness, existential dread, and the horror of their own existence are baseline. But the dragon heritage is what twists the knife. Dragonborn typically carry a legacy of pride, sovereignty, and a fierce connection to their ancestral power.
Suddenly being undead corrupts that. Imagine the shame of your draconic soul, a thing of immense vitality and majesty, being trapped in a decaying vessel. Every instinct to hoard, to dominate, to live eternally is perverted by the state of un-life. The conflict isn't just 'I am a monster,' it's 'I am a dishonored monster, a blasphemy against the very legacy that defines me.' That internal war between a dragon's innate arrogance and an undead's inherent degradation is a fantastic source of angst.
I always go back to that one webnovel where the dragonborn lich-king spends centuries trying to purify his phylactery with dragonfire, not to become alive again, but just to make his eternal existence worthy of his bloodline. That hit different.