Does "I am thirsty to die" make sense?

I am thirsty to die.

The above sentence is more likely to mean "I desperately want to die". It may be considered similar to "I am hungry to die".

If you want to imply "I am dying because of thirst", you can say "I am dying of thirst".

"I am thirsty" stands on its own. A literary or poetic form might be "I thirst to die" but "I am thirsty to die" just sounds awkward.

I am hungry to die and I am thirsty to die both mean the same thing: one has a strong wish to die.

The secondary meanings of hunger and thirst are quite the same.
hunger: Have a strong desire or craving for
thirst: An insistent desire; a craving

Either phrase can be used in the idiomatic sense of a strong desire.
"I am thirsty to die" means the same thing as "I am hungry to die."