First, I think you either need to give up on the idea of the shortcut, or give up on the idea of a road bike. If you want to do that, I guess you could get a vintage road bike with 32mm wide tires, instead of the modern 23mm tires. (skinny) 32s would handle grass OK, as long as there isn't anything too rough in it. If there are thorns, any bike you have will get a flat.

Cyclocross is a good option. It is a ruggedized road bike with slightly bigger, slightly knobby tires, a higher bottom bracket, and more clearance for fenders usually. But they're not cheap, and there won't be a ton of them available used, since they're a pretty recent creation. If you get an older road bike, you can convert it to mostly cyclocross.

What is your budget?

Don't forget flea markets, estate sales, and garage sales. The reason it is so hard to find a good deal on Craigslist is because anyone can surf it while sitting on his butt. The other options take more effort, so you have better odds. You can also look for a bike co-op in your area or ask the police when/where they auction off abandoned bikes from the local bike racks. Maybe check the Classifieds here too.