That actually has the opposite result from what you expect: you wind up with even less food available locally and need higher imports.

When farmers aren't allowed to sell to the worldwide market their sales prices are lower since the pool of potential buyers to bid is smaller. Though this seems fine to buyers at the time, the result is less money for the farmer to invest in future production, and future production is lower than it otherwise would be.

It's similar to the problem with many old-style food-aid programs to poor countries: by just donating food the prices local farmers can get falls, removing their ability to invest in future production. And so the cycle repeats...

"Shortage" seems to be mostly hysteria. "Rising Prices" is real. Some rice breeds not grown in the US actually are getting scarce due to hoarding in Asia, but there is still plenty of rice. Just not cheap rice or the exotic Basmati varieties not grown here.

For many other commodity foods such as wheat and such rising fuel prices are causing farmers to reevaluate what they should grow, and consumer's supply levels and prices are going to fluctuate until prices find the right level to give farmers a reason to produce what consumers want in the desired quantity.