Only 20 players broke par, and just as many were at 76 or worse.
But Woods picked up some late-round momentum in an unlikely spot at the par-3 17th, knocking a 5-iron to 12 feet and burying just the eighth birdie the hole yielded on Friday, producing a huge roar from the crowd.
It was more of a grind for Lowry, who was on the fringes of contention at one-over with four holes to play.
List was in the first group of the day to tee off at 6:45 on the 10th hole. He finished one shot inside the 5-over cut line. His ball on No. 9, his final hole, was in a bunker level with his waste.
Woods watched Brandt Snedeker nail a 5-iron to a few feet at the 17th, but then the wind backed off. Getting some 15,000 people to quiet down proved almost impossible as Woods tried to figure out what to do.
He was at 4-over 144, and happy to be sticking around. He won the 2012 Honda Classic and lost a playoff to Russell Henley two years later, but hasn't made the cut here since.
PGA National continues to be one of the biggest stories of the tournament.
"Another tough day", he said. I was able to do all that today, so that was very pleasing.
"Won't take many more under to win this tournament. Anything around 70 and you're going to be doing well today".More news: F1: Mercedes-AMG Reveals 2018 W09 EQ Power+ Championship Successor
The 14-time major champion made birdies on holes four and nine after an early bogey, but he found water at the par-three 15th.
Tommy Fleetwood, who won the Race to Dubai on the European Tour previous year, scratched out a 68 and was one shot out of the lead along with Webb Simpson (72), Russell Henley (70) and Rory Sabbatini (69).
Fowler shot 40 on the back nine and never got back into the game. "Some of those putts were a little bit bouncy".
On a windy day where Woods has played superbly to get close to the lead, this one was important.
Lovemark backed up his 68 with another under-par round in what were tough conditions at the PGA Tour event.
Nine-time European Tour victor Noren, who fell to former world No. 1 Jason Day in a marathon play-off at Torrey Pines last month, showed no hangover from that loss. The only two-time victor on Tour this season shot an 8-over 78 to miss the cut at 12-over.
Korda, who is the daughter of former tennis player Petr Korda, leads fellow American Brittany Lincicome, who carded a 65 to go 12 under. He birdied his last hole for a 72 and was only one shot off the lead.
"I'm very pleased", said the 14-time Major champion, who had three birdies, a bogey and a double bogey in hard, blustery conditions at PGA National in Palm Beach Gardens, Florida.
He was four shots out of the lead going into the weekend.
Otaegui, from Spain, has one title on the European Tour, a year ago at the Paul Lawrie Match Play Championship.