My passenger side engine mount was just as you described, sagged and all out of shape but still in one piece and holding the engine. I think what happens is when the oil filter is changed the spilled oil from the filter leaks onto the engine mount and degrades the rubber making it spongy. I now cover the mount with rags, stuffing them between the bottom of the filter and top of the mount, to catch any oil that drips out during the filter change. It's a pain getting the rags around the mount when then engine is hot but the effort seems worth it as my mount still looks like new after 5 years.

Hope this helps.