Not a MKI and a MKII, both were MKII (series II) however the 300 was a pre-facelift and my current one is post.
Yes there is a difference in a positive way. The ride is better and it doesn't crash as much over bumps - I can now go over speed bumps without wanting to come to a complete stop and just crawl over them. Not sure if it is only a difference in shocks, springs and bushes may well have been uprated as well.
However it does still crash over an ant in the road and wallow and wander. I am also looking at the Tein CS coilovers.
The wander can be reduced by having a full wheel geometry check. I would do this first because if may solve your problems.
The GS chassis does have a lot of flex in it so chassis braces are probably a good thing to try. Don't think the prolex braces have been tried though. The Toms ones are the best for the GS but are expensive and I am reluctant to buy them without being able to try them first.
Due to the lack of GSs in the UK you will probably be one of the first to try suspension/chassis mods and therefore you will take on a risk of wasting money.