You can diffuse the light source by putting a piece of milk glass in the light path, like at the negative stage in your enlarger.

But the Newton's rings are a function of the contact between the film and the glass, and your light source should, as far as I am informed, not make a difference in this regard.

Questions:
1. Are you using the same film? Some films are more prone to rings than others.
2. Are you using anti-newton-ring glass in your contact printing frame? Or are you using regular glass?
If you are using ANR glass, is the correct side of the glass in contact with the negative?

That's all I can think of. Good luck!