Cenk Tosun then scored a brace for the visitors, the second an 84th-minute victor after Stoke substitute Eric Maxim Choupo-Moting had cancelled out his opener, to leave Lambert's side three points adrift of safety.
Adam, who was sent off for a late and high tackle on Wayne Rooney with the game scoreless, apologised to his team-mates and to Lambert. "I've been in to see [the referee] and he still thinks he's happy with his decision".
"I couldn't ask for any more effort, the way they played being down to 10 men for a long while, I don't think we deserved to lose".
"I couldn't really see it with the weather at the beginning, but I saw it on the monitor and I can understand why Martin's given the red card", the Potters boss said.
"Goals have been a shortfall for us this season and if he's going to be the one that can get into double figures and turn a game like he's done, that would be great". We looked strong and unsafe on the counter, I don't think there was too much trouble in the game, we did lose but we need to start to win.
Speaking straight after the game, Lambert told the media: "The first goal was crucial, and it was offside".
Tosun's stylish second goal, his fourth in three games, earned Everton back-to-back league wins for the first time since December, and the 26-year-old said he was now up to speed after his big-money move from Besiktas in January.
"The thing that disappoints me most is that it is the linesman's side".
"Taking him out of the firing line, giving him the opportunity to get to know the players, do the extra training he's done, we're seeing the full benefits of that now".