Gave it another shot and it spawned me in orbit! I went in to land again as close as I could, and when I got near the surface it was obvious that the planet was flying by under me rather than me flying over it. I started sliding along the surface, so I deployed my landing gear just before it clipped through, just before the shields went down. Now that everything was stable, I decided to go ahead and deploy the SRV...
So then I was flying away from my ship, through the plant at what I assume to be 155m/s or so. Unfortunately, the game still thinks that I'm sitting under the anaconda, even though at last check I was 800km away from it! I eventually clipped back above the surface for a bit in what must have been nearing low orbit. I went downstairs for a few, and when I came back up it looked like I had clipped through the other planet, or that I was close enough now that it was blocking all the light. Still can't fire thrusters or anything, but I did notice that "board ship" was highlighted as an option, so I tried it. The sound effects played, but no luck on being back in the annie. I took a bunch of "starman" selfies and eventually decided that the only thing I could do is log out again (almost dinner time, so daddy has to put his game away).
Sure enough, the servers don't want to allow re-spawning me at my last known location inside the moon, so I am locked out again. I've not received an update from support, so I assume that I was able to log back in because the moon had shifted enough that my last known position was safely in orbit, or that something timed out and I was just placed in orbit of the last known body.
Will post pics in a bit