1) If I'd get a SDD, would I have to run my OS on it too

no, you can definitely use it only for games.

2) Is there anthing else I can try?

Place the game on a HDD of its own, and/or Defragment the hdd.

Having the game on the same drive as the paging and os drive leads to lower performance, i suspect fragmentation following all the recent updates might be a cause of your issues though