CARACAS, Venezuela — The United States is conducting a diplomatic full-court press to try to block Edward J. Snowden, the fugitive American intelligence contractor, from finding refuge in Latin America, where three left-leaning governments that make defying Washington a hallmark of their foreign policies have publicly vowed to take him in. Vice President Joseph R. Biden Jr. took the unusual step of telephoning President Rafael Correa of Ecuador to urge him not to give asylum to Mr. Snowden. Senior State Department officials have also pushed Venezuela, one of the three countries offering to shelter him, with both sides keenly aware that hopes for better ties and an exchange of ambassadors after years of tension could be on the line. And all across the region, American embassies have communicated Washington’s message that letting Mr. Snowden into Latin America, even if he shows up unexpectedly, would have lasting consequences.