The two freewheels could well have large cogs that are different distances from the dropout and derailleur, Did you check the low gear limit screw?

This sounds like a bent derailleur hanger; the extension of the dropout the derailleur bolts onto. (Easily bent any time a bike falls to the right.) Any bike shop can check the alignment of the hanger quickly with a special tool (and straighten it if bent and steel). You can make your own tool for very little money.

Did you take off the Shimano and put on the SunTour? THe SunTour should be less prone to accidental shifts than most Shimanos as the teeth have far fewer cutouts and relieving to promote easy shifting. If you went the other way, to Shimano you may find that derailleurs settings, both limit screws and the lever position are more critical if you don't want unplanned shifts. (I for one actually dislike Shimano FWs and cassettes for friction shifting.)

Ben