Burgess, 34, has previously featured for Darlington and Whitby Town, and joined the club from Scarborough Athletic in 2019.
He said: "I'm buzzing. All done and sorted now so hopefully we can finish off what we started last year."
Blackford, 22, progressed through the youth system at Hartlepool United and made his first-team debut in a 2-1 victory over York City in 2016.
He joined Marske United in 2019 from Shildon.
Blackford added: "I can't wait to play some games and hopefully we get a full season."
He continued: "It will be nice to start training again and hopefully get some friendlies and keep ticking over in the mean time, but it will be good to get some game time."