You are running into normal tolerance issues. The standard lead bullet diameter is 0.358", and that's what Lee seems to have made the ring for.
Many .38/.357 revolvers shoot more accurately with cast bullets that are 0.359" or even .360" for the tumble lube designs. My own 6-cavity .38 WC tumble lube mold throws them .3595" and they shoot groups half the size in my Smith K-38 target revolver of the best commercial match loads. So the size you are getting can be a good thing. Assuming you are using a revolver, just drop them through the chamber throats. Ideal is a close fit from half a thousandth to the lightest bit of drag. For a self-loader, just be sure they chamber. Swaging a cast bullet down a couple thousandths is nothing. In military bore and bullet specs, guns are expected to be able to do that even with jacketed bullets. There are no safety issues with it and though it raises pressure a little, it is within normal pressure tolerances.
As to the Lee ring pushing the bullet in, that's a little annoying. You can probably send Lee your ring and a couple or three of your cast bullets and ask them to open it up a little. You can do it yourself just by splitting a 1/4" dowel and wrapping enough 320 grit wet/dry sandpaper over it to snug into the ring a little, then just turning the ring against it by hand until your bullets start to drop through. That crimp ring may not be as tight on a .357 diameter bullet as you would like, so I'd keep it special for your cast bullets.
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