The odds of Earth or Saturn losing its rotational place around the sun is extremely small. However I will entertain your possibility.
As the two worlds came closer earthquakes and weather patterns would become so wild and extreme most all structures on the planet would be in ruins and billions already dead. The world would begin to heat up the closer we got. If anyone was still alive by the time we reached Saturn's rings we would see the icy shard light up our skies in a spectacle not seen since early Earth. The Ozone layer will collapse and solar radiation will sterilize and kill any life left. Then in a massive fiery explosion we would impact Saturn creating a new storm on the planet that would possibly last millions of years. The Earth would be gone Saturn would remain.