We deal with this problem by subsidizing our agricultural sector and imposing stupid mandates to inflate domestic prices (see ethanol as a perfect example of this). These measures would be outlawed under any "free trade" agreement that included ag products.
I think you missed my principle point. USA agriculture is impossible to out-compete by anyone except for some unique crops such as sugarcane in Brazil or coffee in Columbia.
Overall, if world food markets are opened up on a FAIR and EQUAL tariff basis, American farmers will become the richest farmers in the world. Currently there are so many tariffs and outright bans in other countries for American crops, that is what hurts American farmers. If world markets are available to American farmers, there is zero danger of food price collapse in USA.
As an addendum to my post, please look up the TARIFFS which CURRENTLY EXIST in Canada for USA Agricultural products. As high as 270% tariffs on some items. That is what is killing American farmers, due to lack of export markets.