Former Rotherham, Doncaster and Barnsley man Woods, 29, moves back to England after a season in the Scottish Premiership with Ross County.
“I’m delighted to be here,” Woods told the Shrews’ official website.
“I spoke to the manager a couple of weeks ago and he’s been really quick to get things sorted.
“He’s told me his plans for the season and I spoke to Liam Lawrence as well and he couldn’t have been more positive about the club, so I was really looking forward to getting here.
“It’s a club that’s got a lot of good young players that I’ve seen come through. They play football the right way and get the ball down, and that’s a big factor for me because that’s something I do also.”