The dictionary shows full-time hyphenated as an adverb.
...He works full-time.
...She was there full-time.
As an adjective, it follows the rules: Hyphenate it as a direct adjective; do not hyphenate it when it is not in front of the noun.
...He has a full-time job.
...His job is full time.
Part-time follows these same rules.
Happy punctuating!
Margie