The 26-year-old from Bury has plenty of pedigree at this level, playing for sides such as Nuneaton Borough, Kidderminster Harriers, Southport and Darlington.
At National League level, Dawson has featured for Tranmere Rovers in two separate spells and also Eastleigh.
Speaking to http://www.chesterfc.com, joint-manager Anthony Johnson said: “We’re delighted to welcome Adam to the club. He is a player we’ve tracked for a few years now but never had the opportunity to work with.
“Ability-wise, he’s as good as anyone we’ve seen at this level”.