He was one of seven players released by Aldershot at the end of this season after having a brief loan spell with Dagenham and Redbridge.
Shrews boss Graham Turner said: "I've admired him for some time, I'm really looking forward to working with him."
Morgan had been placed on the transfer list following a message he posted on social networking website Twitter.
BBC Radio Shropshire understands he turned down a chance to join newly promoted AFC Wimbledon.
Source: BBC Sport
Source: BBC Sport