The Category 1 hurricane was expected to strengthen slightly before making landfall later in the day, bringing with it up to 15 inches of rain.
The storm was packing sustained winds near 80 miles per hour as it headed toward the U.S.-Mexico border and the heavily populated Rio Grande Valley.
A hurricane warning was in effect for the coast of Texas from Brownsville to Corpus Christi and in Mexico from Rio San Fernando northward.
The National Weather Service in Brownsville said conditions were favorable for tornadoes in deep South Texas and the adjacent coastal waters. Forecasters predicted Dolly could dump up to 20 inches of rain and bring coastal storm surge flooding up to 6 feet above normal high tide levels.
In Mexico, officials planned to evacuate 23,000 people to government shelters. Texas officials were urging residents to move away from the Rio Grande levees.
At local stores there was a run on plywood to protect windows, but as CBS Early Show weather anchor Dave Price reported, Dolly's most serious threat is water, not wind.
Texas officials urged residents to move away from the levees because if Dolly continues to follow the same path as 1967's Hurricane Beulah, "the levees are not going to hold that much water," said Cameron County Emergency Management Coordinator Johnny Cavazos.
The first bands of rain began to pass over South Padre Island and Reynosa, Mexico Tuesday afternoon and the surf continued to get rougher.
Tropical storm warnings were issued for areas adjacent to the hurricane zone, and Gov. Rick Perry declared 14 South Texas counties disasters, allowing state resources to be used to send equipment and emergency workers to areas in the storm's path.
CBS affiliate KGBT reported that Perry had also put the state's military force on alert - making 1,200 men and women with the Army National Guard and Air and Texas National Guards ready for deployment orders in response to Dolly.
"Currently we have soldiers based in Austin, San Antonio, Houston," Col. Bill Meehan of the Texas Army National Guard Center in Austin told KBGT. "And we have two air units that are prepared to launch; one out of Austin, one out of San Antonio."
The storm, combined with levees that have deteriorated in the 41 years since Beulah swept up the Rio Grande, pose a major flooding threat to low-lying counties along the border. Beulah spawned more than 100 tornadoes across Texas and dumped 36 inches of rain in some parts of South Texas, killing 58 people and causing more than $1 billion damage.
"We could have a triple-decker problem here," Cavazos told a meeting of more than 100 county and local officials Tuesday. "We believe that those (levees) will be breached if it continues on the same track. So please stay away from those levees."
Around Brownsville, levees protect the historical downtown as well as preserved buildings that were formerly part of Fort Brown on the University of Texas at Brownsville campus. Outside the city, agricultural land dominates the banks of the Rio Grande, but thousands of people live in low-lying colonias, often poor subdivisions built without water and sewer utilities.
The International Boundary and Water Commission, which operates a series of levees, dams and floodways in the lower Rio Grande Valley, put its personnel on standby alert.