His side beat Cheshire on Sunday to retain the title they won last year.
However, with crowds at the final falling, there is speculation that the Rugby Football Union could take the game to a smaller ground in future.
"Hopefully they don't take it away from the stadium, because young players want to play at this stadium," said Dawe.
"It all depends on the RFU, doesn't it? How they keep this competition going, how they keep it alive."
Cornwall have now reached the showpiece game in each of the last four years.
The final of the competition, also known as the Bill Beaumont Cup, has been played at Twickenham since 1984.
"It is a problem, it's not perfect by any stretch of the imagination," Dawe added to BBC Sport.
"All you can do is the best with what you're dealt and don't look at it too gloomily, but who knows what the future holds."

Victoria Street in Victoria flooded at about 18:00 BST on Saturday and the water gushed into several businesses including the Curzon Cinema, which had to be evacuated.
Fire crews then helped some people in the venue cross the flooded pavement.
Thames Water said a large pipe burst damaging the road but no customers were without water on Sunday.
A spokesman for the company said: "Our team is on site arranging emergency repairs.
"We apologise for any inconvenience caused and we will fix the pipe as soon as possible."
The road remained closed between Palace Street and Buckingham Gate on Sunday morning and several buses in the busy area were on diversion.
On its website Curzon Cinemas Victoria said: "Due to a burst water main, Curzon Victoria has had to close tonight (Saturday 6th June).
"We apologise for any inconvenience caused."

Italy, Wales' first opponent in the 2017 tournament, have finished bottom of the table 11 times, and Georgia and Romania have been pushing to be included in the top tier.
But the 30-year-old, set to win his 46th cap in Rome, believes that would be a premature move.
"I think it's a debate to be had in a couple of years time," he said.
Italy's captain and talisman Sergio Parisse reacted angrily when the suggestion was put to him during the 2016 tournament.
And Six Nations chief executive John Feehan has said relegation will not be introduced in the short to medium term and dismissed criticism of Italy as unfair.
Owens expects the Azzuri to pose a serious challenge in Rome, where Wales have lost twice and believes that if relegation is eventually introduced, the teams and players will deal with the issue.
"I don't think any team goes into the competition to finish bottom," he said.
"If [relegation] comes, it comes and the focus wouldn't change because no side goes into the Six Nations wanting to win the wooden spoon."
Italy coach Conor O'Shea has made five changes from the team which lost against Tonga in November for the game against Wales.
It is also the first time Wales have faced the Italians since the appointment of their new coaching team.
"It's a tough away game, especially Italy first up," he added.
"They'll be wanting to build on a good autumn campaign after beating South Africa and they'll have been disappointed losing the last game against Tonga.
"But they've had time now with Conor O'Shea, Mike Catt and Brendan Venter in camp.
"We know hard it's going to be out there and it's all about getting a good performance first up."
Cornwall coach Graham Dawe hopes the County Championship final is not moved from Twickenham as part of a potential revamp of the competition.
[NEXT_CONCEPT]
Cinema-goers were carried to safety by firefighters after they became stranded when a water main burst in London.
[NEXT_CONCEPT]
Wales hooker Ken Owens has backed the decision not to introduce relegation from the Six Nations.