Also any tips for practicing softer Ukemi in general would be appreciated
Generally the more of your body to make contact with the mat at the same time the better, that way the force is spread out over a much larger surface. But every teacher / orginazation has a different take on ukemi. Mine take take comes from my years in the USAF-WR/Birankai & Judo. But when it comes to koshinage, that is one of those techniques where uke is pretty much completly at nages mercy. It gets easier over time and you will get used to it.
IMO the quickest way to get used to hard falls would be to take a month of Judo