Biggins had been offered a new deal by Doncaster but opted to leave following their failure to win promotion after a storming late-season run.
He played an integral role in Rovers’ 2-0 play-off semi-final first-leg win at Crewe, scoring his side’s second goal.
But the former Barrow and Fleetwood man immediately went off with a hamstring injury and missed the second leg, which Donny lost 2-0 before being beaten on penalties.
“When I spoke to him it was clear he wants the opportunity – and has the hunger – to prove himself in the division once again,” said Town head coach Paul Hurst.
“What stands out is the way he makes the box. Hopefully, he will bring some goals and assists to the team.”
Shrewsbury were the lowest scorers in the EFL last season, netting 35 goals in 46 games.
