What's the difference between "morar" and "viver"?
They mean the same... viver also means "to keep" or continue the same way. Ele vive triste = he is always sad. A torneira vive pingando = the tap keeps leaking.
Is it correct to think of using "morar" in the physical location sense, such as "I live in an apartment", and using "viver" in a physical location or state sense, such as "I live in an apartment" or "I live in perpetual anguish"?
Yes. That's a good thought. I added some other points above.
In this case, "live" would be similar to "survive." This phrase also reminds me of a song by the Mexican pop band Maná (Vivir sin Aire). Look it up.