Is there a way to move files to multiple folders in single Folder Action?
You can use a shell script like this:

for f; do
  case $f in
    *.mp3|*.m4a) mv -n "$f" ~/Music;;
    *.jpg) mv -n "$f" ~/Pictures;;
    *) mv -n "$f" ~/Documents;;
  esac
done


mv -n skips a file when a file with the same name exists in the target.