Shaw will spend the rest of the season on loan at Bristol before his new contract officially starts.
The 23-year-old has already had several loan spells at Gloucestershire and is leaving in search of more regular cricket.
Yorkshire are well-stocked in the seam bowling department and Shaw has found first-team opportunities difficult to come by.
“It wasn’t an easy decision to make but sometimes you’ve got to do these things to get the best out of yourself,” said Shaw, who turned down a contract extension at Emerald Headingley.
“The main aim for me now will be to target playing regular first-team cricket in all three forms.”