“Morosely,” of course, is an adverb that is based on the adjective “morose.” “Morose” can be defined, as in the dictionary link below, as “sullen and gloomy” or “very serious, unhappy, and quiet.” A person who is morose is unhappy, but they are not unhappy in a loud and angry way. A morose person will tend to withdraw into him or herself and be relatively uncommunicative. A person who does something “morosely,” then, does it in a quiet and gloomy way.
The word “morosely” is used three times in Of Mice and Men (according to the search function on my Kindle). Each time, it is applied to George. Very early in the book (p.4 in my paperback copy) George “stared morosely at the water.” Later (p. 12), he “stared morosely at the fire.” Finally, on p. 23, he “fell morosely silent.” In all of these cases, he seems to be angry at Lennie. These incidents all show how irritable George can be with Lennie and how he seems to need to lash out at Lennie to feel better about the fact that he is not achieving the things that he would like to in his life.