9. I'm thinking of buying a present for my son's birthday.

(Does 'for my birthday' modify 'bying' and is it an adverbial clause as well? Or is it an adjective phrase modifying 'present'?))

10. I need two dollars for the bus.
(Does 'for the bus' modify 'need' and is it an adverbial clause as well? Or is it an adverbial phrase modifying 'two dollars'.)

9. "for my son's birthday" is serving as an adjective phrase modifying "present," yes. It's a phrase because there's no subject/verb in it, and it's an adjective phrase because it's modifying/describing a noun.

10. In this sentence the prepositional phrase seems more adverbial because it's telling why it's needed; it's not describing the bus. (PS. Adverbs don't modify nouns, which is what "dollars" is.)

Adverbs answer one of these four questions: How? When? Where? and Why?
http://www.chompchomp.com/terms/adverb.htm