David Woodley quarterbacked the LSU Tigers in college. He led the team to a 34-10 victory over Wake Forest University in the Tangerine Bowl in his last game as a Senior, taking home the game's MVP award. After graduation, he was drafted in the eighth round of the 1980 NFL Draft by the Miami Dolphins.
In 1980, Woodley went 6-5 as a starter, taking over for Bob Griese and Don Strock. He rebounded from an embarrassing first game as a starter, tossing two interceptions and completing 11 of 28 passes for 48 yards in a 34-0 shutout against the New England Patriots to eventually win the team's MVP award. He led the Dolphins to a week 10 victory against the Los Angeles Rams, completing 61 percent of his passes for 161 yards and three touchdowns. He threw for a total of 14 touchdowns against 17 interceptions with a 54 percent completion rate and 1,850 passing yards. Unfortunately, the Dolphins missed the playoffs that season, finishing at 8-8.
In 1981, Woodley started 15 games for the Dolphins, posting an 11-3-1 record and leading the team back to the playoffs again. He totalled a career high 2,470 yards on 191 for 366 passing, collecting 12 touchdowns and 13 interceptions in the process. He started the Dolphins playoff game in the Orange Bowl against the San Diego Chargers, going 2-for-5 with an interception and letting the Dolphins fall behind 24-0 before being pulled in favor of Don Strock. Miami would come back, but ultimately fall short, 41-38 in one of the best games ever played.
In 1982, Woodley posted a 7-2 record in a strike shortened season, leading the Dolphins to their first Super Bowl appearance in nine seasons. The Dolphins would eventually fall behind a 166 yard effort by Redskins running back, John Riggins, 27-17.
Woodley opened the 1983 season as the starter for Miami. Midway through a week five loss to the New Orleans Saints, Dan Marino was inserted in his place. Marino would start for the remainder of the season.
For Miami, Woodley totalled a 27-12-1 record with 34 touchdowns and 42 interceptions, passing for 5,928 yards on 508-of-961 passing. He passed away in 2003 due to complications of kidney and liver failure.