He said rain continues to fall and flooding in certain areas may last another week.
More than 20 people are reported dead, and large parts of the city of Houston are under water.
The storm has now moved on to neighbouring Louisiana, where flash-flood warnings have been issued.
Governor Abbott said Texas has carried out more than 8,500 rescues, and more than 32,000 people remain in shelters throughout the state.
Another 10,000 members of the National Guard are said to be on their way, joining the 14,000 already deployed to tackle the disaster.
The storm, which previously reached hurricane status, has dumped record rainfall in the region.
However, meteorologists say much drier weather is forecast for Texas in the days ahead.
Authorities and volunteers are still battling to rescue stranded people.
The sheriff's office in Harris County has confirmed a family of six - two great-grandparents and four children - drowned while trying to flee the floods in a van.
Among them was 16-year-old Devy Saldivar, who posted an anxious Facebook message about the extreme weather on Sunday: "No sleep & anxiety, please let this pass sooner."
"Our worst fears have been realised," said Sheriff Ed Gonzalez, breaking the news.
The driver, the son of the elderly couple, was able to escape and was found clinging to a tree, screaming for help, he said.
Authorities are reminding people not to drive - many of the casualties happened on the roads as people underestimated the power of the water. On social media, officials are repeating their mantra "Turn around, don't drown" to stop people heading out into difficult situations.
Meanwhile, in Louisiana, the storm landed in the south-west corner of the state, around Cameron, in the early hours of Wednesday. It is predicted to travel in a north-easterly direction, weakening as it goes.
The National Weather Service said heavy rainfall was expected from Louisiana to Kentucky over the next three days.
Flash-flood warnings remain in effect for parts of southern Louisiana and southern Mississippi.
New Orleans - which was devastated by 2005's Hurricane Katrina - is not in the storm's direct path, according to forecasters.
In Houston, which is the country's fourth most populous city, a night-time curfew has been implemented to prevent looting, while homes remain abandoned.
The curfew runs from 00:00-05:00 local time (05:00-10:00 GMT) for an indefinite period. Relief volunteers, first responders, and those going to and from work are exempt.
Photo from Houston Chronicle