I looked in the 1900 census for John Albertson in New Jersey.The only John Albertson I found born in December 1897 was born in Pennsylvania and living in Philadelphia.His parents, John and Mary J., were born in December 1853 and November 1857.He had two sisters, Jennie, born June 1880 and Cora born November 1889.Both sisters and the parents were born in New Jersey as were the parents of the parents. According to the census Mary had 5 children with 3 living.
I tried searching for the other children you listed and could not find them in any census.
The New Jersey Archives should have records that can help you.
If you can, I suggest you go there yourself.
If that is not possible go to this web site and click on Genealogical Holdings.
You will find instructions for ordering the birth certificate of Jennie and Cora Albertson.The fee is $10.00 for each record.This should tell you the name of the parents including the maiden name of the mother.
According to the 1900 census John Anderson and his wife, Mary J. were married for 25 years.I checked the marriage index at the above site for a John Anderson married about 1875 but could not find one.With the information from the birth certificates you should be able to search the marriage database using the name of the mother.