Dawson was part of the England squad that won the 2019 Cricket World Cup, however, he did not play in any matches during the tournament.

After appearing regularly for Hampshire Second XI and Wiltshire in 2006, Dawson was selected for England under-19s tour of Malaysia in 2006/07.

In 2012, Dawson's bowling began to become more prominent, as he picked up 26 wickets for Hampshire in the County Championship, averaging 32.

In 2013, Dawson scored over 1,000 first-class runs for the first time in his career, although he was less effective with the ball, taking just eleven Championship wickets.

In October 2018, Dawson was named in the squad for the Comilla Victorians team, following the draft for the 2018–19 Bangladesh Premier League.

He topped the Professional Cricketers' Association rankings for Domestic MVP and the County Championship Player of the Year.

[6] In 2024, Dawson again achieved the feat of a century and ten wickets in a match when scoring 104 not out and taking 10/99 against Lancashire.

[9][10][11] Dawson won the County Championship Player of the Year at the 2024 Cricket Writers' Club Awards.

In November 2016, Dawson was named in England's Test squad for the final two matches of the series against India.

On 6 July 2017, Dawson was called up to the England squad for the South Africa Test series and was selected to play as the second spinner on the side.

[23][24] In September 2021, Dawson was named as one of three travelling reserves in England's squad for the 2021 ICC Men's T20 World Cup, though he played no part in the tournament.