Mawkishly is the adverb of mawkish: it tells you how someone is showing the feeling. They are doing it in a way that is too thick, too sweet, and a bit embarrassing — speaking mawkishly, smiling mawkishly, writing mawkishly about a small goodbye as if the world were ending. The meaning is the same as mawkish; only the job in the sentence has changed. It is still a critic's word, not everyday chat. Though actually, once you know mawkish, you already know this one. English just adds -ly to turn the adjective into a "how" word.