So, I finally figured out something crucial about eyelets and yarn overs - the yarn is supposed to go over the needle counter clockwise - the same way as it does for a knit stich so that the little loop sits on the needle with its front leg forward. That way, you get nice holes for eyelets. Otherwise, you have an extra stitch but it's twisted so there is no hole. The good news is that I finally figured this out. I've been doing this wrong at least a good chunk of the time so sometimes I had a nice eylet but mostly, I didn't.
The bad news is that I figured it out in the middle of the project. I was torn - should I start doing it right or continue doing it wrong? Having figured out how to do it right, I was found myself doing it right naturally so that's what I did. I'm not sure that was the right decision. It's a lacey shawl and I fear it's going to look not good. So do I frog? Keep going and hope it turns out? I've spent hours and hours on this project - I don't want to do it all over again, but I've spent hours and hours on it and so I really want it to turn out well.