Close to engagement to one man: I touch ring at one point (7)
I believe the answer is:
tangent
'i touch ring at one point' is the definition.
(a tangent touches a circle or ring)
'close to engagement to one man' is the wordplay.
'close to' is a charade indicator (letters next to each other).
'engagement' becomes 'tan' (I can't explain this - if you can you should give a lot more credence to this answer).
'to one man' becomes 'gent' (gent is a kind of man).
'tan'+'gent' is 'TANGENT'.
(Other definitions for tangent that I've seen before include "Straight line touching a circle" , "Line grazing another" , "Line touching circumference" , "What people often go off at!" , "Particular line" .)