Here's what I've tried so far. I ended up filling a bobbin with matching poly thread then using embroidery thread on the top. The both seem pretty comparable in how thin they are.
It's better, but it's still not quite what I want. My next try will be to try a pattern that came with my machine, rather than one I downloaded for free. I want to make sure it's not a dud pattern. Then I'm going to try another stabilizer/interfacing. I wonder if mine isn't thick enough.
If I'm still having trouble...I might just give up. lol
Thanks for everyone's great suggestion. Now I'll put them all to use, 1 at a time.