The 28-year-old links back up with Robins' head coach Steve Cunningham, with the pair initially working together during their time at Buxton.
Dawson is a product of Leeds United's academy and made his first team debut in 2013 in a 2-1 defeat against Derby County.
After departing the club for Rotherham in 2016, he joined Norwegian Club Viking FK on loan, and then went on to sign a contract with the then Northern Premier League club Scarborough Athletic.
The former Wales U21 international had subsequent spells at Bradford (Park Avenue), Grantham and a return to Scarborough, before moving to Buxton in 2020.
Dawson will go into contention for Ashton United's opening fixture of the 2023/24 season away to Workington AFC on Saturday.
Photo credit: Jason Chadwick