Apparently there's a whole branch of physics devoted to the study of what's known as "Aeolian vibration" and is of great interest to grid / transmission line operators.
Usually demonstrated in one one of two ways apparently; "gallop" or "flutter". Gallop also known as 'conductor dancing' according to Wikipedia and is a low frequency (about 1 Hz) and high in amplitude oscillation - hence the dancing bit. As opposed to flutter, which is also resonant but of high(er) frequency (around 10 Hz) and low amplitude.
Your issue sounds, pardon the pun, more like "flutter" - which in larger structures and long transmission lines, apparently can be cured by fitting so called mass dampers (I always think of the millennium bridge in London) or Stockbridge dampers.
Sounds like in your case some easing of the tension in the catenary/drop, or maintaining the return twists as noted above will have a similar dampening effect on the wire.