I run mine out of gas. I use Sta-bil all summer, and I have a product from Penzoil that is a stabilizer/winterizer that I start using in the gas come September, so that's usually what I run the mower with when I run it dry. I also wash the underside off, and scrape what I can with a plastic putty knife to not scrape up the surface any more than it is.
After the engine cools, I'll pull the spark plug, squirt a bit of engine oil in the cylinder, put the plug back in (important step as the next step will get oil all over the top of the deck if you forget, and you'll only forget once...), and pull the cord a couple of times to distribute the oil, then try to park the cylinder on the compression stroke and put the mower away.
I change the oil in the spring. The acids thing is on the back of my mind, but I got 14 years out of the last mower and never changed the oil in the fall, and the engine was still going strong when I got rid of it. YMMV, of course.
In the spring, I change the oil, pull the plug and give it a bit more oil, put in a new plug (if needed), put on a new air filter, gas it up with fresh gas with Sta-bil, and let 'er rip.