Are you sure those are true co-axial cables? Most RCA patchcords like that, that are flexible, actually contain 2 inner wires that are twisted loosely around down the cable (not twisted pair, but definitely not coaxial).
In addition to grounding them, switch to heavy duty video-quality 75 ohm coaxial cable (RG59 or RG6, shouldn't matter for short distance), not twisted cable such as in those flexible RCA cables. At the very least you'll end up with a ~ 20 guage center "data" wire (22 gauge on RG59) instead of the tiny stranded ~24-26 gauge wire I've seen in most flexible RCA cables.
You will find your picture improving significantly. It's how the higher end monitors with BNC connectors work.
Or, if you dont want to convert to rigit Coax, at the least, trim the length of these down to the shortest you can deal with, AND place a ferrite bead around both ends of the cable to reduce interference entering.. That's why VGA cables for computers have big protusions on each end, since they have such small stranded wire inside (though a common shield).
Remember.. Original arcade wires going to monitors were typically 20 guage wire... thick enough to withstand some interference...