After laying down the required cards that complete a phase, does the player continue to draw cards or does he play only the cards that remain in his hand?
Yes, they keep drawing
A player still does all the steps of a turn, regardless of whether or not they have completed their phase (i.e. Draw, lay down a phase if they haven't already and can, optionally add any additional card(s) to a laid-down phase, discard).
This allows more time for other players to finish their own phase while still making 'finished' players try and scramble to empty their hands on completed phases to avoid getting points.