I think you are definitely on the right track.A little more research is needed but with what you already know and what I can add here, I think you have enough clues to lead you to your answers.
First, my information agrees with yours in that Rev. Johnson Adams b. 1785 and wife Elizabeth had a son William K. Adams who married Nancy Burchfield December 14, 1837 in Blount Co. TN.My records show they had a son Johnson/Johnston K. Adams.
Here are some of the census records I have located on William K. Adams and Nancy Burchfield Adams:
1850 Polk Co. TN Census
page 511, dwelling 897, stamped apge 256
William K. Adams, age 37, farmer, Value of RE $300, b. TN
Nancy E., age 30, b. NC
Elizabeth, age 11, b. TN
Margaret M., age 9, b. TN
Johnston, age 6, b. TN
Robert, age 4, b. TN
Levi, age 2, b. TN
1860 Macon Co. NC census, enumerated 11 July 1860
Page 269(stamped), dwelling 573
William K. Adams, age 48, B. Preacher, RE$200, PP$189, b. TN
Nancy E., age 41, b. Burke Co. NC
Johnston K., age 16, b. TN
Robt. B., age 14, b. TN
Levi L., age 11, b. TN
Matilda A., age 9, b. TN
Thomas W., age 7, b. Cherokee Co. NC
William J., age 5, b. Cherokee Co. NC
Lydda A., age 2, b. TN
Dan'l E., age 5/12, b. Cherokee Co. NC
This is where my data on this line ends for the most part.
I did find that the above Johnston K. Adams had a cousin named Johnson Adams who was born about the same time (1844-1845), and I do not know which of these two men is shown in the census records below.
Is this the Johnson Adams, son of John Preston Adams (who was son of Rev. Johnson Adams), that is shown in 1870 and 1880 Cherokee Co. NC census below?Or is it instead Johnston Adams, son of Wm. K. Adams who married Jane DeWeese 31 Jan 1867 in Macon Co. NC per IGI?It appears that the name of the wife and the age of the firstborn are a fit to be the latter, but this in and of itself does not prove which of these men this really is since the names Johnson/Johnston are used interchangably and they were cousins born with a year of each other. More research needed.
1870 Cherokee Co. NC census, Pg 294, household 107
Johnson Adams, age 26 b. NC
Jane age 22 b. NC
Bunyon age 2 b. NC
1880 Cherokee Co. NC census, household 347, Murphy, NC
Johnson Adams, age 37, b. TN, father b. TN, mother b. TN
Jane, age 33, b. NC, father b. NC, mother b. NC
Bunyan, son, age 12, b. NC, father b. TN, mother b. NC
Ella, 10, b. TN, father b. TN, mother b. NC
Adda, 8, b. TN, father b. TN, mother b. NC
Nettie, 5, b. TN, father b. TN, mother b. NC
Arthur, 1, b. TN, father b. TN, mother b. NC
Oscar, 1, b. TN, father b. TN, mother b. NC
Chrystal, if the above census record is that of your ancestor Johnston K. Adams, then you have the names of his children born by 1880. You didn't say when your Horace Ivan was born.If I assume we have the right family above, since he does not appear in the 1880 census record I would then assume it was after 1880.Does any of this fit with your facts?
Do you know if they stayed in Cherokee Co. NC? If so, I can probably find them in the 1900 census on Ancestry.com.
Hope this helps you.I'd like to extend my data on this line if you find it connects and are willing to share your data.