Phantom flush

I did a Google search on this topic, and everything I found focused on replacing the flapper, but this toilet does not HAVE a flapper, so here I am. First, I had a small but steady leak into the bowl, with no phantom flushing, for several days. My husband replaced the fill valve, and when that didn't fix it he replaced the flush valve gasket, which stopped the leak… and that's when the phantom flushing started, although there is a small but nonzero possibility that it started after he replaced the fill valve, but I didn't notice it until he replaced the flush valve gasket the next day. To be clear about the phantom flushing: it is a few seconds of that starting-to-flush sound, not a complete flush. By the time I can run into the bathroom, any motion of water into or out of the bowl has ceased, so I don't know what is actually happening, just that it's making the noise. Any ideas on what to try next would be greatly appreciated!! :D