But because of a four-and-a-half-hour storm delay the pair had managed only five holes of their third rounds and the final day will be a marathon affair for all the leaders.
McDowell, joint third overnight, birdied the first and then chipped in at the short third to move to 11 under par.
Winner of the WGC-Cadillac Championship in Miami in March, first round pacesetter Watney also took advantage of the soft greens on the resumption of play and birdied the first two holes.
That left halfway leader David Toms, who missed a string of makeable putts, one behind rather than one in front and he shared third with Steve Stricker.
England's Luke Donald had his first bogey of the week when he failed to get up and down from just off the fifth green, but he had birdied the long second and so was eight under with German Martin Kaymer, who like him is shooting for the World Number One spot as well as the title.
Kaymer burst into a share of first place with four opening birdies and then another at the seventh.
But the German, who could go back to World Number One ahead of the absent Lee Westwood with a top-two finish, bogeyed the next three before coming back with yet another birdie.
Londoner Brian Davis was right in the mix after birdies on the first two, but he went back to six under with bogeys on the fifth and seventh.
Phil Mickelson, like Donald requiring a win to go top of the rankings, reached six under when he eagled the 16th for the second time in the week and then almost aced the 17th.
But in the gathering gloom he bogeyed the last for a 69 and five under aggregate.
Scot Martin Laird, his playing partner for the third day running, managed only a 76 and so was down on two over, while Justin Rose had slipped back to two under and Ian Poulter was one further back.