Old relative an escort, embarrassingly (8)
I believe the answer is:
ancestor
'old relative' is the definition.
(relative from past generations)
'an escort embarrassingly' is the wordplay.
'embarrassingly' indicates an anagram.
'escort' with letters rearranged gives 'cestor'.
'an'+'cestor'='ANCESTOR'
(Other definitions for ancestor that I've seen before include "One of one's forebears" , "Old member of the family -no traces (anag)" , "Enactors (anag)" , "Adam was our first one" , "Predecessor" .)