Lucygrace, I just tracked a line of my family tree back to Beesley Island as well. And since your posting was only a couple of weeks ago maybe we can get in touch. My first look research shows Francis Beasley, b. 1550 (assumed England) > George, b. 1586, Lancashire, England > Robert, b. 1601, Lancashire, England Died 1693, North Carolina > Sarah Beasley b. 1680, North Carolina.
(There may be a Robert Jr. I missed, or her father was 79 when she was born. Her mother, Sarah Pettit Loadman Beasley was born in 1644, age 44 when the daughter was born, so it could be right.)
I came across a reference to Solomon Beasley, b. 1701, Beesley Island, Craven, North Carolina, which is how I found this line.
This is my first look at this line so far back so take it all with a grain of salt. I got here by searching backwards from Effie Bass Miller, my great grandmother, died and buried in Kissimmee, Florida.