Welham, 24, played alongside Cooke for Hull KR in 2007, and has spent the past two years with the Hawks, scoring 15 tries in 33 appearances.
He said: “Cookie sold the club really well to me when we spoke about the possibility of joining Doncaster.
“I felt coming here was a good opportunity for me and the club’s prospects look promising after having a great season back in the Championship.
“They’ve got a good structure in place and are heading in the right direction and I know a lot of their Hull based players who speak highly of the club.”
Welham has also previously played for Hull, Featherstone and Dewsbury.
Cooke, who succeeds Tony Millers as Dons head coach, said: “Liam’s a good lad with a great attitude and he’s exactly the type of player I want at the club.
“I’ve played in the same team as him in the past and I’ve also played against him this season and he’s a very good signing for us.
“I believe he has the right kind of qualities that will make him a firm favourite with our fans at the Keepmoat.”