My first thought was about the Motor mounts, they probably need replacing very soon, that rocking is not good.
Have you done a compression check? Lots of people are going to start screaming about doing a leak down yes, but a compression check will get you some decent results quickly and either find a problem or eliminate a source of the problem. If the engine was rebuilt several years ago, and there have been multiple carbs on it with no improvement it is probably something else. Do you get any water coming out of the tail pipe?.
