Those key midfielders are suspended for Saturday night's game, with both clubs hoping to end three-game losing streaks.
A poor start to the season for Dallas (3-6-3) took a turn for the worse Thursday when MLS announced a three-game ban for Shea, who kicked the ball at an assistant referee in a 2-1 loss to Columbus last Saturday.
Shea, tied for the team lead with three goals while leading the club with 24 shots and 10 shots on goal, posted an apology on Twitter. Dallas is 0-4-2 since beating expansion Montreal 2-1 on April 14.
Adu was assessed two yellow cards in the first half of the Union's 3-2 home loss to New York on Sunday. The second one was called for simulating a foul in the penalty box.
"When I went down I wasn't complaining to the ref or complaining to anyone," Adu told the Union's official website. "I just wanted to get right back up and the next thing I know the red card comes out."
His absence comes after Philadelphia (2-6-1) scored two goals in a match for the first time. The Union's seven goals in nine games represents the league's third-worst ratio ahead of winless Toronto FC and Chivas USA.
That attack will face a makeshift defense, as Dallas is missing two defensive starters. Ugo Ihemelu will miss his third straight game in the aftermath of a concussion while Zach Loyd is suspended for receiving a red card last week in Columbus.
The club has conceded 18 goals in 12 games, tied with Los Angeles for the second-worst ratio in MLS ahead of Toronto. Dallas has also been issued a league-high four red cards.
"We've had some bad luck with some late goals and bad luck keeping the players on the field," coach Schellas Hyndman told Dallas' official website. "You can say that's bad luck, discipline or intelligence, but those are things that are in our control."
Philadelphia, which ranks right behind Dallas with three red cards, traded defender Danny Califf to Chivas for midfielder Michael Lahoud and allocation money Thursday. Califf started four games, but would have been the odd man out with Gabriel Farfan set to return Saturday after a three-game suspension.
"We have enough speed right now," coach Petr Nowak said. "The trade is going to give us another option as well."
Lionard Pajoy had both goals for the Union on Sunday after scoring once in a 2-1 exhibition victory over German club Schalke 04 on May 9. Those games featured the inclusion of fellow forward Danny Mwanga in the lineup to give Pajoy support.
Dallas is 2-0-2 in the all-time series with Philadelphia, taking both home matchups.