I also have a designer SE. Hope you get it worked out. I have used Solvy Heavy weight and vilene (a WSS woven type)both with success. I've use size 14 and 12 needles. Dealers tend to suggest changing your needle after about 8 hours of use. I am surprised how much that can make a difference in the appearance. I have used RA, Sulky and Madeira threads. My dealer suggested NOT using Coats and Clarks.
Oh that is interesting regarding the thread. So was the Solvy heavy weight easy to use in your se? I was thinking about buying some or badgemaster. I've only used the vilene.
Hope the thread is your problem & some different will fix it. *
I see you have an great answer - so I wish you well with FSL and will leave you with a flower.
When I started I had the same experience. I asked my dealer and took the embroidery with me. He said that is was the yarn, that was a little bit stretchy. I could use that yarn for all embroidery but not for FSL. After I bought a new yarn it went well. Hope this helps you.