"Good" new road bikes at $300, no. Sub 20 pounds, no.
However, there are a number of entry-level choices in the $200-$300 range online which will serve you well. They'll weigh generally mid-20 pounds, have 7-speed freewheels, downtube or stem shifters or microshift integrated shifters at the high end of your budget.
If you look hard enough, know your stuff and have the mechanical knowledge to rehabilitate a used bike you might be able to find a decent used bike on Craigs List that weighs 20 pounds. I suspect it would require replacing parts of it. Personally in your shoes I'd adjust my expectations to an entry level bike and look at Bikesdirect. Any of those will last for years, having standard (entry level) modern parts that will be easy to repair or replace, if that ever becomes necessary.
Or you can spend weeks studying craigslist, researching the different models as they come up, pounce on a project bike and dive in on the wrenching. There's nothing wrong with that and maybe you wind up with a better bike. But there's something to be said for ordering a brand new one, delivered in 3-5 days and after minimal assembly you're on the road instead of watching CL ads.