I've just been watching your YouTube video and noticed you selected RF output then used the RF to AV converter plug to the RCA inputs of your TV. I personally set the switch to video and I screw on a RF-RCA adapter and connect a RCA video lead from the camera to the television so I get the video signal directly. I suggest you try doing that and see how the picture turns out, if it turns out good then it's an RF issue, otherwise the problem lies elsewhere.
Secondly the sync must be set to internal unless the camera is hooked up to a CCU or and SEG which can be externally synchronized, but from what I seen in your video it's set to internal so that's not the issue.
Also some information on this camera from my knowledge of them. These cameras were made around 1970 to I think about 1974 when the AVC-3250 was introduced, so your camera is from the early 70s. The image pickup tube used for these cameras are 2/3 inch vidicon tubes which were very common in B&W cameras during the 70s. The Sony AVC-3200/10s were very popular in schools and colleges for shooting events and educational productions.
You can find more info on the similar AVC-3200 on Labguysworld site
http://www.labguysworld.com/ on this page
http://www.labguysworld.com/Sony_AVC-3200.htm . Send him an email too if you need some technical advice as he knows a lot about vidicon cameras and has built one from scratch.