Full-time or full time, part-time or part time?

It might depend where your 'here' is:

full-time

adj: for the entire time appropriate to an activity

a full-time job; a full-time student

adv: full time: on a full-time basis:

he works full time

Compare part-time; full-timer n Collins English Dictionary – Complete and Unabridged © HarperCollins Publishers 1991 . . .2003

full-time

adj.: working or operating the customary number of hours in each day, week, or month. Compare part-time.

adv. : on a full-time basis. [1895–1900]

[Bob works full-time]

full′-tim′er, n. Random House Kernerman Webster's College Dictionary . . .

[quotes tidied slightly & augmented but not significantly changed]

You have to check individual acceptable styles individually (so Lumberjack's reference, while interesting and accurate, may not help here) and be prepared for different 'rulings' from different authorities.

Incidentally, the above two dictionaries are consistent in their treatment of part-time / part time. AHD doesn't include the adverb polyseme.

Usually people are more unsure over whether it's full-time or fulltime. The correct is hyphenated. As an adjective/adverb (in this case, of working), it is hyphenated. This is the way these words are used most often.