Willingly isn't an issue, or even an expectation....it is a responsibility and legally you have to - like it or not.

She willingly helped her friend move apartments, despite it being a long and tiring task.

He willingly volunteered to help with the project.

I am willingly answering your question.

The man willingly went into the city to see could he help after the disaster.

She willingly went to the police station for questioning about the crime.

It would be an adverb. If you said "I willingly gave up my last cookie," the word "willingly" would be an adverb because it describes a verb. Adverbs typically modify verbs, adjectives, or other adverbs.