I'm having the exact same problem, only with the rectangular model! It rained here for two days, and Starlink went out. Said it was disconnected at the dish. I went to check the connection, and there was a tiny bit of water on the plug. I brought the whole thing in, dish and all, and dried it with a hair dryer. I let it sit for a while, then plugged it back in. Still said disconnected at the dish. I was using the 150' cable at that time, so I tried my old 75' cable. It worked that time, so I put Starlink back outside, this time on the roof of my camper instead of the yard.
After two days, the same thing happened - it showed disconnected at the dish. I unplugged the cable and found tiny water droplets on the plug. That cable had been perfectly dry, stored inside, and there hadn't been any rain after I started using the 75'. The moisture HAD to have come from inside the dish.
Like the OP, I have followed all the directions and cared for my Starlink system according to the recommendations. There's no damage I can see. It was working perfectly before the rain.
