They reached 11 under but trail world number four Jordan Spieth, who moved up from fifth with six birdies in his 67.
Phil Mickelson, one shot off the halfway lead, dropped six shots in his 75 and is six shots back in joint 18th.
US Ryder Cup player Patrick Reed had a hole-in-one with a five-iron at 16, the 20th ace on the PGA Tour this season.
Mickelson claimed a share of the lead after a birdie at his first hole but unravelled with some untidy play, registering a double bogey at the fifth.
"I didn't play well today," he said. "The last three bogeys hurt. I fought all day without really playing well to keep it at even par and then the last three holes got me."
Playing alongside the five-time major champion was qualifier Austin Cook, a 24-year-old from Arkansas.
He led the tournament before a double bogey at the 15th, but birdied 17 in a round of 70 and is one shot behind Spieth, who is hoping for a good performance before the Masters at Augusta, which starts on Thursday.
"Today was as comfortable as I've ever been with the lead on a weekend," Spieth said. "I wanted to get into contention as much as I could prior to the Masters, to have as much experience as I could to limit those nerves.
"Then I'll take even more confidence into next week."
Casey, 37, a three-time Ryder Cup veteran, won the KLM Open on the European Tour last year and was second at the Northern Trust Open in February.
The world number 48 had four birdies in five holes on his front nine.
Day, 27, who is ranked 1,045 and plays on the second-tier web.com Tour, did not drop a shot in round three and had four birdies on the back nine.
Englishmen Paul Casey and Kelvin Day carded rounds of 68 to earn a share of sixth place, three shots adrift, after round three of the Houston Open.