James Magee is in the 1840 census for Washington County, and is shown as employing free people of color as well as owning slaves, so it's more likely that the apprentice bond for David Pearce would have been issued in Washington County. But, if Rhoda was a white woman who was living in someone else's household, then she won't be found in the 1840 census. David is listed as black in the 1850 census, and then as mulatto in the 1860 one. I haven't yet located him in 1870 - do you know how he was listed in that and later census records?
One thing I did find was a Freedman's Bank Record entry at ancestry.com for James Henry Sparrow. The date is May 15, 1873, and he states he was born in Washington, NC (no notation as to if this is Washington County, or Washington, Beaufort County), and he's 27 years old. He lists his father as Canady Sparrow and his mother as Rhoda Pearce, both deceased. He names siblings: Washington Keyes, Daniel Moore and Robert Moore. The combination of having a mother named Rhoda Pearce and brothers with the surname of Moore definitely caught my attention, but with there being no mention of David Pearce...I'm not sure. I did find a Daniel and Robert Moore in Plymouth in the 1880 census, as boarders in a household, and working as hands in a saw mill, but I have not yet found them in 1870 (or James Henry Sparrow).
Is there anything in the family lore about why or how David ended up in Beaufort County?