I think "excited" is definitely an adjective in the first sentence, and most likely an adjective in both sentences. It looks like some people have argued that it must be a verb in the second because of the phrase starting with by, but I think they're wrong. My viewpoint would be that it is technically indeterminate in the second sentence.
"Excited" passes several adjective tests
I would base my judgment on the tests mentioned by BillJ in his answer to the following ELU question: Is “running” a gerund or a participial adjective?
First-off, the very test. Very can't be used by itself as a pre-posed modifier of verbs (we can't say *"This very excites me"; we have to say something like "This excites me very much") but it usually can be used as a pre-posed modifier of most adjectives (we can say "I'm very happy").
"I was very excited about my new job" is clearly OK, so "excited" is acting like an adjective here, not like a verb. Strictly speaking, this doesn't prove that it's an adjective in "I was excited about my new job", but my intuition is that these sentences have the same structure.
"I was very excited by my new job." This construction is not as common, but people do say things like this, and it sounds grammatical to me. So I would say "excited by _" can also be an adjective phrase, although when not preceded by "very", it could technically be a verbal form (passive voice) instead.
The other tests BillJ mentioned:
can it be the complement of complex-intransitive verbs like "become": again, "I became (very) excited about my new job" passes with flying colors, and "I became (very) excited by my new job" seems fully grammatical to me.
can it be the complement of complex-transitive verbs like "feel": again, "I felt excited about my new job" is good, "I felt excited by my new job" is... kind of funny, actually. So maybe "excited by ___" is not so good as an adjective phrase after all.
Overall, "excited about my new job" is clearly an adjective phrase, while "excited by my new job" seems to only sometimes be treated as an adjective phrase (or only by some speakers). Of course, the other analysis of "I was excited by my job" is that it is a passive-voice version of "My new job excited me", which is something that can be said in English (although it sounds a bit weird to me). That's why I said "I was excited by my new job" is technically indeterminate on its own.
Other viewpoints about the significance of "by"
In the course of trying to find the answer to the linked ELU question, I came across this web page from "the Internet Grammar of English": Participial Adjectives.
This resource seems to be linguistically oriented, as it has connections to the UCL Survey of English Usage, so I expected it to give a correct description of how to distinguish adjectives and participles.
However, there is one part that seems questionable to me. It says
The presence of a by-agent phrase (by your behaviour, by my reaction) indicates that the -ed form is verbal.
It also says this actually overrides the very test for adjective-hood:
very can sometimes be supplied in both the adjectival and the verbal constructions:
I was embarrassed I was embarrassed by your behaviour
I was very embarrassed I was very embarrassed by your behavior
She was surprised She was surprised by my reaction
She was very surprised She was very surprised by my reaction
However, I don't know if -ed forms followed by by-agent phrases really are commonly accepted to be an exception to the very test.
Reasons I doubt the by-rule
To me, it seems like all of these sentences have adjectival phrases. I don't see why a by-agent phrase would be a certain indicator of a verb in this context, since it seems this type of by-phrase can also occur after other parts of speech, for example after abstract nouns denoting actions as in "the destruction of the library by Caesar". (It's true that in this case the object also seems to be required to be present in a phrase starting with of, but I don't see why that would be relevant either way.)
Also, "embarrassed by your behaviour" doesn't seem to behave very much like a verb, as we can say things like "She became embarrassed by your behaviour", "She felt embarrassed by your behaviour", "She stopped being embarrassed by your behavior."
Another reason to doubt that "by" is significant: the un-rule
The supposed by-rule also seems to clash with another test for adjective-hood that I found in "Another look at participles and adjectives in the English DP", by Tibor Laczkó, who cites unpublished work by Joan Bresnan that identifies un-prefixation (excluding the separate verbal prefix un-) as something that can only occur to adjectives. Obviously, we can say things like "She was unsurprised by my reaction", so this would also indicate that by-agent phrases are not an infallible indicator of verbal forms. (It isn't proof that surprised is an adjective in "She was surprised by my reaction", but I think it's proof that a following by-agent phrase doesn't make it impossible for surprised to be an adjective.)
However, I don't know if the un-test for adjective-hood is widely accepted; maybe some people would say that un- can be prefixed to the participle forms of certain verbs. Unfitting and unbecoming can take direct objects (although this is rather archaic nowadays; both of them have alternative constructions with prepositions like unfitting to and unbecoming of), but despite this, I think they are generally classified as adjectives rather than as participles.
A relevant article
Discussion in "Adjectival participles revisited", by Artemis Alexiadou, Berit Gehrke and Florian Schäfer (2012):
3.1 By-phrases and event-related modification in adjectival passives
The standard claim for English is that adjectival passives are incompatible with by-phrases, and to our knowledge there is little discussion on other event-related modifiers in English adjectival passives.
The generality of this claim has been challenged recently for many languages, for which
it has been shown that event-related modifiers are available [...]
English data from McIntyre (2011) and Bruening (2012):
(19) a. The dictator remained unsupported/underestimated by the warlords.
b. Former investigator says he remains disturbed by what he saw at baby murder scene.
c. No longer does Tim Thomas appear trained by Tim Hortons.
d. There are others who I would call saints more than theologians since they seem taught by God more than by men.
e. Once one monkey discovered a new food-washing method, very soon the whole tribe used the method, untaught by the original simian.
f. Steve Jobs’ birthday doesn’t go unnoticed by spammers.
➩ There exist adjectival passives with by-phrases also in English and German.
➩ But these are restricted compared to by-phrases in the verbal passive, or compared to by-phrases in Greek adjectival passives.