The Crows were denied a go-ahead goal against Sydney with 70 seconds remaining in their one-point loss which tipped them from the finals race on Saturday night.
A Ben Keays kick was ruled a behind when the goal umpire incorrectly deemed the ball had brushed the post, and the AFL has apologised to Adelaide for the error.
"It's a very crucial time of the year, and a game and a moment, for us," Dawson told reporters on Monday.
"We want to show our disappointment.
"But we also don't want that umpire to feel like he has cost us the year.
"Because we know there have been plenty of games where we probably should have won and have been in these positions that we have lost and cost ourselves through our own error.
"There's nothing that can be done. I watched it a couple of times but there's no point continually watching it and getting frustrated by it."
The defeat ruined any chance of the Crows breaking a finals drought stretching back to 2018.
John Longmire, coach the seventh-placed Swans who are now assured a finals spot, sympathised.
"It's important to acknowledge how disappointed I'd be if I was Matthew Nicks," Longmire told reporters.
"There's no question that you get disappointed when those kinds of things happen.
"They don't happen very often but if you're on the wrong side of it, you get very disappointed.
"The AFL have acknowledged that. It's disappointing for the AFL, disappointing for the umpire involved. We feel for him as well.
"Footy can be a cruel game sometimes."
The AFL has stood down the goal umpire, who didn't seek a video review, for the rest of the season.
"That is what the technology has been brought in for, to use in cases and scenarios like that where the decision, the umpire is not sure," Dawson said.
"He obviously thought it was a point and he stuck with the decision.
"Obviously for us it's disappointing because we would have loved to have used the technology in that scenario.
"But it's just one of those things.
"It's human error. Don't take it out on the goal umpire because at the end of the day we didn't play the footy we wanted to."
The Crows are in 13th spot, having lost five games by six points or less this season.
Two of those were against table-topping Collingwood.Â
After a two-point defeat to the Magpies on June 25, the AFL also apologised to the Crows for a field umpire missing a free kick to Dawson in the dying seconds which would have offered a relatively simple shot at goal.
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