Sinking
[sinking into a black desk chair in his office in suburban Atlanta ] means =he begin to work

Question

Answer

"Sinking" does not, on its own, imply doing anything else. If the passage followed a phone call giving him bad news, "sinking" would imply a feeling of helpnessness or resignation.  It really depends on the context.

Answer

Meaning:
To slowly move your body into a sitting or lying position, in a relaxed or tired way.
Reference:
Cambridge Dictionaries Online