Hello, Earlene, I'd start researching Pulaski County, MO, where 1840 William Moore (b. 1761-1770) and a younger William Moore (per will his son) which was listed in Pulaski County Probate microfilm. By 1849 Laclede County was formed from part of Pulaski and your Rachel (Moore) Hudson was listed 1850 census which also listed same census page Nathaniel R. Rodgers as a neighbor. Rodgers was listed on the William Moore probate records 1846 of Pulaski County. The probate records for the elder William Moore are listed as follows:
Pulaski County, Missouri: William Moore, Probate Records (familysearch.org digitized microfilm, Letters of Administration, v.2, 1833-1852)
PP. 231/298/320 Moore, William 1846 (index)
(no Hudson listed)
A bond of Twenty Thousand dollars placed on Joseph J. Owen, Executor of the Estate of William Moore and Elizabeth Moore, E.H. Harrell, J.A. Rayl, A. Hamor, Samuel Mitchell, E.J. Christison, Bland N. Ballard, Elisha Christison, John Morgan, R. W. Rodgers, Cyrus Colley, and J. N. Davis as Securities dated this 11th day of March 1846... the condition of the bond is that Joseph J. Owen Executor of the Estate of William Moore with the will annexed shall faithfully administer the Estate...
p. 232-234,JosephJ. Owen, Executor of the Estate of William Moore, stated the Heirs of Wm Moore to be Elizabeth Ann Moore, wife and four children; Margaret Jane Moore, George J. Moore, Mari M. Moore, and William H. H. Moore, all of the County of Pulaski in the State of Missouri... 16th March 1846, J.J. Owen, Robt A. Hardin, Clk
Pulaski, Missouri; will and testament of William Moore, deceased, was Exhibited, proved, and recorded in the office of Clerk of Court of Pulaski County, a copy herein annexed...
Joseph J. Owen was appointed Executor... signed 16 March 1846, Robt A. Hardin, Clk, Certified by Robert J. Hardin, 17 March 1846
Wm Moore, late of Pulaski County died Testate and being that his property may be lost, destroyed or diminished in value if speedy care is not taken, the court appoints W. H. Clary (Public Administrator) to administer with the will annexed the goods chattel which was of the said Wm Moore at the time of his death... 26 March 1849, Andy Anderson, Clk
To all persons to whom these presents shall come greeting, whereas Wm Moore, Archibald McDonald, Jacob Casebolt, and R. A. Hardin, died testate and having at the time of their death having property in this state which may be lost, destroyed, or diminished in value if care not be taken... we do hereby appoint Blan N. Ballard, administrator..23 May 1857, J. A. Anderson, Clerk. State of Missouri, County of Pulaski, J. A. Anderson, Clerk of the County Court in and of the County and State aforesaid do hereby certify the above
Letters of Administration this 23rd day of May A.D. 1857. A. Anderson, Clk