After a frustratingly bloody day, Mooncup UK FAQ came to the rescue (much more helpful than the one for Diva). They suggested clamping your pelvic floor muscles around the cup to get a good seal, so I tried that after rotating and while doing the usual pushing in-out motion. After the clamping, everything felt a bit more secure. So far, not a spot, and I'm thrilled. I hope this works!
Who else has a pretty regular leakage problem? Both my mom and I do, and we've been using them since last summer.