Joe Garner's double sandwiched by a superb strike from Paul Gallagher put the hosts 3-0 up with little over an hour gone at Deepdale before Vale fought back late on.
Gavin Tomlin and young substitute Jordan Hugill pulled goals back for the visitors as Adams was left to rue what might have been.
"Over the course of the game, if you look at the amount of chances we created, we certainly feel we could have got something out of the game," said Adams.
"I was reasonably happy for the first 42 minutes. I thought we frustrated them. But the goal on half-time was frustrating because we didn't stop the cross and we didn't attack it.
"With the personnel we had available to us, we've done reasonably well.
"Sometimes you can argue that you're 3-0 down, you've got nothing to play for so it's easy to play, but I thought we stuck at it.
"We got back into the game. We've created chances and I think we can feel ourselves a little unfortunate.
"But the key moments, for me, showed we're a little flaky with our defending at times and we can't afford to do that away from home against the better sides."