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Mich is the direct object, however folgen always takes the dative case, no matter what. Other examples of dative verbs are gefallen, helfen, danken, antworten, and so on. So no, it has to be "Ein Mann folgt mir.
Also, your second sentence is correct, but folgen and verfolgen don't have the exact same meaning. Folgen simply means to follow, while verfolgen is more like to track or pursue.