Walshaw, 36, has been the Seadogs top scorer in both full seasons since his arrival from Farsley Celtic and had initially pledged his future to the club, while Taylor recently went on trial with a club in the Scottish Premiership.
Dawson became the third player to make the move on Wednesday afternoon.
Manager Darren Kelly said: “I got a message from Wally this afternoon and he told me that he was leaving for personal reasons.
"I had a telephone conversation with Tommy and he told me that he had decided to join Buxton.
"I had driven to Leeds to meet him on Friday and discuss things because he is a player that has been brilliant since I took over as manager. I'm gutted to see him go.
"Tommy has made his decision on his future now, so we have to move on.
"I received a text from Chris and he told me he was leaving for Buxton.
"He is a player that had a lot of injuries last season and again, we move on."
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