Forecasters warned of up to 138cm of rain that could produce flooding and breach levees in the heavily populated Rio Grande Valley. Thunderstorms were attributed to Dolly as far away as Houston, 644km up the Texas coastline.
In Mexico, fields were filling with water, palm trees were bent over in the wind and beaches were closed to the public.
Maria Miguel, 102, and seven family members fled their wooden shack in the fishing community of Higuerilla and spent the night at a convention center-turned-shelter in Matamoros. "I don't know if my poor house will withstand the rain and wind," Miguel said.
Mexican soldiers made a last-minute attempt to rescue people at the mouth of the Rio Grande. The soldiers battled storm-charged waves in an inflatable raft to rescue at least one family trapped in their home, while others further inland were still refusing to go to government shelters, said Matamoros spokeswoman Leticia Montalvo.
"These are people who did not want to leave, and now they are in trouble," Montalvo said.
On Texas' South Padre Island, an apartment complex roof partially collapsed early Wednesday. Residents said they did not believe anybody was injured.
The causeway linking the island to the mainland remained closed early Wednesday.
Dan Quandt, a spokesman for South Padre Island emergency operations, said winds were picking up to around 50 mph (80 kph) and were expected to increase later Wednesday morning. He said there was a steady rain falling, but no reports of flooding. A sign on a hotel blew off, but no one was injured and it did not pose a hazard, he said.
Power was knocked out to more than 13,000 customers in Cameron County, at the tip of Texas where Brownsville is located, utility company AEP Texas said. Power also was out on South Padre Island.
In Brownsville, palm trees leaned and small debris was strewn across the all-but empty streets. The windows and doors of shops were boarded up with plywood and most businesses - including gas stations - were closed. At one gas station, workers were pelted by horizontal rain as they scrambled to lock pumps and close down.
Cities and counties in the Rio Grande Valley were bracing for massive flooding and levee breaks. Tornadoes were also possible near the storm's path.
Local officials urged residents to move away from the Rio Grande 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.
Charles Hoskins, deputy emergency management officer for Cameron County, said there were nearly 2,000 people in six shelters in the county late Tuesday.
In Hidalgo County, a little bit farther inland, six shelters holding about 900 people were open, said Cari Lambrecht, a county spokeswoman. She said people living in low-lying areas were encouraged to come to shelters.
In Mexico, Tamaulipas Gov. Eugenio Hernandez said officials planned to evacuate 23,000 people to government shelters in Matamoros, Soto La Marina and San Fernando.
The U.S. Census Bureau said that based on Dolly's projected path, about 1.5 million Texans could feel the storm's effects.
Gov. Rick Perry declared 14 south Texas counties disaster areas, allowing state resources to be used to send equipment and emergency workers to areas in the storm's path.
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 (91 centimeters) of rain in some parts of south Texas, killing 58 people and causing more than $1 billion damage.
Around Brownsville, levees protect the historic 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.
Immigration and Customs Enforcement was evacuating its Port Isabel Detention Center, said spokeswoman Nina Pruneda. Fewer than 1,000 people were being sent to other detention centers in Texas.
In the Gulf of Mexico, Shell Oil evacuated workers from oil rigs, but said it didn't expect production to be affected. It also secured wells and shut down production in the Rio Grande Valley, where it primarily deals in natural gas.
Mexico's state-run oil company, Petroleos Mexicanos, said it had evacuated 66 workers from an oil platform off the coast of the port city of Tampico. Pemex said in a statement that it had readied a team and the resources needed in case of damage to oil installations in the region.
