Local resident Walker posted three birdies in five holes from the 14th, capped by back-to-back birdies at the finish.
"It's a tough golf course," said Walker, who won the Sony Open in Hawaii in January and is aiming for his fifth PGA Tour title.
"You definitely want to find a way to extend the lead, but you have to be smart about it."
Walker took a bogey at the par-four 12th when his tee shot buried into soft sand in a bunker left of the fairway and had to use his hands to dig around the left lip to find the ball.
Spieth, who won his last PGA event two weeks ago at the Valspar Championship, shot a rollercoaster 71 that included six birdies, three bogeys and a double-bogey at the par-three 16th to stand second at five-under 211.
"It was up and down," said Spieth, who was three-under on the front nine. "Then I didn't play the smartest shots, was being a little too aggressive and it bit me."
A two-shot swing at the par-five 14th where Spieth bogeyed gave Walker some breathing room and the margin was extended when Spieth launched his tee shot at the par-three 16th well right of the grandstand and took a double-bogey.
FedExCup champion Billy Horschel birdied the last three holes for a 71 that put him at three-under, one shot ahead of a group of five at 214 that included two-time Texas Open winner Zach Johnson.
Charley Hoffman, who trailed Walker by one after 36 holes, soared to a wind-blown seven-over 43 on the front nine with three bogeys and two double-bogeys and shot 79 to finish the day 12 strokes behind Walker.
Phil Mickelson saw the highs and lows on the course in shooting a 74 to stand at level par 216.
The big left-hander struggled with his short game in shooting a front nine of 40 then nearly holed out for double-eagle from 258 yards on the par-five 14th and made an eagle.
(Reporting by Larry Fine in New York; Editing by Gene Cherry/Greg Stutchbury)
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