Why do capacitors across my FETs minimize ringing/distortion?
I'm not entirely sure which picture is for which configuration but adding the capacitors is not a good idea because they will reduce the efficiency of your switching circuit. The transformer produces the ringing because it is inductive and it has self capacitance - together, when the FETs switch you will get a resonant tuned circuit and ringing will occur.

I'd do nothing other than try to improve your transformer. Maybe you should give details of this. There are also other effects that can appear to cause ringing namely your o-scope earth point not being close to your measurement point and getting an induced voltage in that loop from the switching currents. This commonly happens. Make sure your measurement technique is sound.