The dictionary form is "맴돌다" meaning "to be/hang around someone"
계속 내 주변를 맴도는 듯해
☞ You seem to keep hanging around me or You seem to be still around me.
More Information,
"맴돌다" is a 'ㄹ' irregular verb like "만들다(to make)" or "살다(to live)"
(verb stem ends with 'ㄹ', it disappears when it's followed by 'ㄴ', 'ㅂ', 'ㅅ')
맴돌다 ☞ 맴도는 듯해 (맴돌 + ㄴ 듯해 ☞ 맴도 + ㄴ 듯해)
맴돌다 ☞ 맴돕니다(맴돌 + ㅂ니다 ☞ 맴도 + ㅂ니다)
만들다 ☞ 만드는 듯해 / 만듭니다
살다 ☞ 사는 듯해 / 삽니다
\^o^/