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You should avoid visiting others without an appointment.
Literal
Promise without other-person [obj-を] visit [nominalizer-こと][topic-は] avoid-[must-なければいけません].
約束なしに ('without an appointment') uses なし ('absence, without') as a nominal version of the negation — なしに is adverbial 'without.' 他人 ('other people, strangers') is distinct from the generic 人 ('person'), implying non-family non-intimates. 避ける ('avoid') + こと nominalizes the action.