Fulton and fellow Bairn Stephen Kingsley signed for the Swans yesterday.
Fulton said: "I'm absolutely delighted. "I think I've done better than expected since I came down here in January, and I can't wait to get started for pre-season.
"Everything has happened pretty quickly but I didn't need any persuading to sign and I'm really looking forward to the future.
"I just want to get a good pre-season under my belt now and see where it takes me.
"It's nice to have had a break but I just want to get back out on the pitch again and pick up where I left off."
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