@Aces_N_Eights & @Bluey: I really can’t express sufficient gratitude for your advice. I think you’ve diagnosed my problem accurately, and I also found a video that I think covers the same problem: https://www.youtube.com/watch?v=E1zwQvL9mt8
He talks about the wrong way of swinging (leading with the bottom) “creating a figure 8,” which is exactly what’s happening with my rope, though I wasn’t able to describe it nearly as helpfully. He gives a cue–thinking about looking at your watch as you swing it around in front–that seems to be the same idea as your image of “wiping your brow.”
I may take you up on the offer of posting a video if I’m still struggling with the swing after practicing today. Again: thank all of you for your advice on this!