It was not Welle's intention to cause a nationwide panic, but that's what he did.
The infinitive phrase is to cause a nationwide panic.
Is this phrase a noun, adjective, or adverb?
I think a noun because it restates what his intention was not?
I agree.
Welles had designed the play to sound like an actual broadcast with flash news bulletins.
infinitive phrase to sound like an actual broadcast with flash news bulletin,
I think this phrase is either an adverb that says how it was designed or a adjective that describes the play?
I vote for adverb, describing designed.
People tuning in were shocked to hear that giants from Mars had landed in New Jersey.
Infinitive phrase to hear that giants from Mars had landed in New Jersey.
This one I do not know but it describes shocked...
You are correct! The infinitive phrase "to cause a nationwide panic" is functioning as a noun in this sentence. It is acting as the direct object of the verb "intention."
To determine the function of the phrase, it helps to identify what role it plays within the sentence. In this case, "to cause a nationwide panic" answers the question "what?" after the verb "intention." So, it is effectively restating what his intention was not. As a result, it functions as a noun phrase.