A component cable will make a big difference in video quality, it's quite noticeable on my 42" 1080p TV, but the problem you're describing may still remain. The fact that things change when anything moves suggest that you maybe having an issue with your TV's video processing. Check to see if your TV has a game mode that you can enable for the input that you've connected your PlayStation 2 to. Video processing generally doesn't work well with games, and as it can cause a fair bit of lag, most TVs have a mode that disables it.
It shouldn't be too hard to find a multi-system component component cable for sale pretty cheap that supports the PlayStation 2. Also a PlayStation 3 component cable will work, as both consoles use the same AV connector.