Wow, these replies have so much good advice for me. Thank you, thank you, thank you to each one who responded. It's also good to know that there are others who suffer from "can't-stipple-itis."
I have studied each response and am going to try all of these various approaches. With all of this good advice, I feel like I do indeed have many personal stippling coaches.
I definitely want to master this aspect of quilting and just hadn't thought about the fact that it is a skill that must be developed (at least for folks like me), and must be practiced continuously. I'm off to start practicing with pencil and paper -- so many people suggested that. I plan to get one of Leah Day's Supreme Sliders (drat that woman for making stippling and everything else about quilting look so easy!). My sewing machine is currently sitting on top of a table, but I plan to look for a cabinet or table that will allow it to be recessed so the top of the sewing surface is level with the table. That will be a present to myself when I celebrate my birthday at the Houston Quilt Festival next month.