Vale took the lead after just 37 seconds when Chris Neal's long ball evaded everybody and Doug Loft pounced to easily slot his eighth goal of the season into an empty net.
Ben Williamson scored the second after 73 minutes after heading in from two yards out.
Jordan Clarke headed in substitute Mark Marshall's free-kick with 10 minutes to go before Callum Wilson's solo effort in the 90th minute completed the comeback.
And the former Coventry manager was unhappy with his side's defending to allow his former emplyers to grab a last-minute point.
Adams said: "I think we've got to feel disappointed with what we've done for their two goals, we've not defended the set-piece well and it's quite clearly a foul on Jennison Myrie-Williams for the second.
"It's a real shame as we put a real shift in. Up until the first goal we conceded I was delighted.
"We gradually got into the game and Ben Williamson and Tom Pope were a handful for them.
"Up until the last 10 minutes we were comfortable. It's unbelievable defending, and when the players see that last 10 minutes I'll think they will be embarrassed. They've switched off and it's frustrating for our supporters."