がち is used to describe a tendency or habit, often negative. Here, it means "busy people tend to forget meals.

気味 is used to express a slight tendency or feeling. Here, it means "I feel slightly like I have a cold."

げ is used to describe an appearance or manner that suggests a certain feeling. Here, it means "he had a sad-looking face.

「気味で」 follows the stem of a verb or a noun. Here, 「疲れ」 is the stem of 「疲れる」, making it the correct choice.
Translation: "Lately, work has been busy, and I’m feeling a little tired."

気味 (ぎみ) is used to indicate a slight tendency or feeling. In this case, it means "he seems slightly tired.