Randall is a Reds Academy product who first joined the club at the age of six.
The 20-year-old right-back made his first-team debut in the League Cup clash with Bournemouth in October and also featured in the recent 6-1 mauling of Southampton.
Fulton is the Merseyside club's U-21s goalkeeper and has twice been named on the bench for the first team.
Randall told Liverpoolfc.com: "I'm obviously made up to sign a new deal, giving me the chance to stay here for longer, keep working, looking to improve under the manager.
"I'm delighted and very grateful. Since the first day you come in, that's the goal. You want to get into the first team, be involved and around it.
"It's a dream come true to be in the situation I am. To sign a contract today, I'm very pleased and grateful for the chance."
Fulton added: "I'm absolutely buzzing and it's great to have a bit more security and stay at a club like this and I will try to kick on from here now.
"I want to keep going and try to progress as much as I can. I've got the best chance of that here working with Simon [Mignolet], Adam [Bogdan] and John [Achterberg] and the rest of the players. The standard here is brilliant and it makes me better every day."