Coulson has been a regular feature at the Flamingo Land Stadium since joining from St Johnstone, and is new boss Jono Greening's first recruit since taking charge at the club.
Greening said: "I'm delighted to have him onboard for the year again. Scarborough is in his heart, he is a top player and a top pro.
"When I was at York running the 18s, he was a massive presence for them and I played against him when he was a whippersnapper at Barnsley. It has been a joy to watch him over the years and I'm very pleased to have him as captain of the club.
"He's not just a creative player who can score goals, he is one that puts a shift in for the team, which is exactly what I want."
On Watson, the manager added: "I'm delighted Ryan is staying. He's a very good, technical player, who is good from set-pieces, experienced and good in the changing room.
"He is somebody that will be a very big player this season and I'm excited to work with him."
The club also announced the appointment of new player-assistant manager Michael Ingham from Tadcaster Albion.
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