I'm still struggling with holding on too tight.
But at the end of the day, Craig Weirman recommended that I completely change how I'm holding myself on with my legs, and put more weight on the front part of the bike. We covered this a bit during a one-on-one session first thing in the next day. I went out to practice that for the rest of the day. This really appears to work, although I'm struggling to adjust to how different the bike moves under me. But when I do it right, I'm really light/easy on the bars. So I'm committed to this change. But it's hard work.