The name is spelled DAVISON in census records -- not DAVIDSON.

In looking at the 1880 census, W.J. Bridges is listed as the step-son of C.B. Davison so, yes, I would say that Rachel was married to a Bridges first. W.J. Bridges is 16 and Julia DAVISON is 15. I don't see a Mary E. Bridges in the household.

The census also says W.J.'s father was born in KY while C.B. Davison was born in S.C.

Here's the family in the 1870 census and it's obvious by Rachel's age that she must be a second wife to C.B. (Calvin) DAVISON because she's too young to have given birth to the two oldest boys, John and William, unless she was 12 yrs. old.

Source Citation: Year: 1870
Census Place: Benton, Polk, Missouri
Roll M593_800; Page: 9

Calvin DAVISON, 40, Farmer, TN
Rachel DAVISON, 32, MO
John M DAVISON, 19, AR
William J DAVISON, 18, AR
Mary F DAVISON, 10, MO
Nancy J DAVISON, 7, MO
Julia I DAVISON, 5, MO
Martha E DAVISON, 3, MO
Josephine G DAVISON, 1, MO
Wm J Bridges, 7, MO
Jack Williams, 19, MO

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An online family tree at ancestry.com by bstinebaugh@comcast.net shows the following:

Name: Calvin Bracket DAVISON
Sex: M
Birth: 1 MAR 1830 in Maury City, Tennessee
Death: 6 DEC 1903 in Fate, Rockwall County, Texas
Death: 6 DEC 1903 in Rockwell City, Texas

Marriage 1: 10 MAY 1849 in Polk County, Missouri
Married: Matilda Jane Glover b: 28 NOV 1830 in Missouri
Children --
William Johnathon DAVISON b: 30 AUG 1851
Mary Francis DAVISON b: 22 JAN 1859 in Marshfield, Webster County, MO
Nancy Jane DAVISON b: 16 JAN 1863
---------------------------
There seems to be a 2nd marriage posted by jwoodw@aol.com

Marriage 2: 3 JUL 1851 in Polk Cty, Mo
Matilda Anglin b. abt. 1830 in Tn
Children:
George DAVISON
John DAVISON b: ABT 1851 in Ar
William DAVISON b: ABT 1852 in Ar
Mary DAVISON b: ABT 1859 in Mo
-------------------------------------------

Marriage 3: 6 SEP 1864 in Polk City, Polk County, MO
Married: Rachel J. (Stinebaugh) Bridges b: 19 JAN 1838 in New Hope, Lincoln County, MO
Children --
Julia Isabel DAVISON b: 28 JUL 1865 in Missouri
Martha Ellen DAVISON b: 29 SEP 1866 in Polk City, Polk County, MO
Josephus Grant DAVISON b: 7 MAR 1869 in MO
Dora Ann DAVISON b: 25 APR 1871 in Fate Rockwall County, TX
Emma Viola DAVISON b: 11 MAR 1875 in TX
Clarissa Polon DAVISON b: 29 MAR 1877 in TX
Jacob Calvin DAVISON b: 16 AUG 1881 in TX

Calvin is the son of Brackett DAVISON b: ABT 1803
and Delilah Hardison b: ABT 1805; d. Abt 1885

Rachel is the daughter of Jacob Stinebaugh , The Younger b: 28 NOV 1794 in Wilson County, TN
and Nancy Cannon b: 29 APR 1800 in Hopkinsville, Christian County, KY

Marriage 1: Homer Bridges b: 1826 in Kentucky
Married: 1859
1 Child - William Jasper Bridges b: 27 FEB 1863

I hope you can sort out which children belong to which mother. It's a little confusing.