When you find yourself saying "help! I don't know anything! I can't find anything!" it can help to write out everything that you do know, and how you know it, and then to put all those bits and pieces into a timeline about the person. Imagine you have an assignment to write a short biographical sketch, as if you were writing the person's obituary. Make an outline of the things you know and would like to include. It's okay to also add in some notes about the things you don't know (when and where he died, when and where he married, and so on).
Then after you have your list, you can look at what you've assembled and see what questions you might be able to ask and answer. Make a 'wishlist' of records you would like to have but haven't found yet, like an obituary.
Now you can use the research guides to work the individual problems. If I'm reading your question correctly, you think your grandfather Allie T. Russell was raised by people who were not his birth parents. Congratulations! You can have twice as much fun as people who only have relatives who were born and grew up with their birth parents. You can research his adoptive family and his birth family.
Important resource: RootsWeb's Guide to Tracing Family Trees, Guide No. 31: Adoption and Orphans Research
Your grandfather's birth record is going to be a challenge no matter what because he was born in 1892, before many parts of the United States had statewide birth registrations.
The section about Birth Records in the Family Search Wiki article on North Carolina vital records tells us:
Statewide registration of births and deaths began in 1913
and was generally complied with by 1920. In some cities record keeping
began earlier. For example, Raleigh began recording births in 1890 and
deaths in 1885. Counties where the births and deaths occur keep a
duplicate copy of the information they send to the state office.
Tennessee is about the same, with statewide recording in 1914 and general compliance in the late 1920s. FamilySearch Wiki: Tennessee Vital Records.
From this we know that any record of your grandfather's birth will have to be found at the county level. Sometimes people who were born before the beginning of statewide registration apply for a birth certificate later in life -- these are called delayed birth records. Family Search's article for North Carolina says:
Due to Social Security requirements and other federal acts, many
people needed proof of birth. If no birth record was available, they
could go to the county where they were born and file a delayed
registration of birth. These may be found in county offices of the
Register of Deeds. These records may list births from the 1870s to the
1960s. They usually give the exact date of birth, town or city of
birth, and often the full names of both parents, as well as the volume
and page.
FamilySearch's article on Tennessee Vital Records says that Tennessee began issuing delayed birth certificates in 1935.
FamilySearch's article about North Carolina vital records also says:
Birth records usually give the name and sex of the child; the names,
birthplaces, and ages of the parents (with the mother’s maiden name);
the occupation of the father; and the number of children born to the
mother. Birth records of adopted children may give the birth parents
but have frequently been amended to show only the adoptive parents. A
year-by-year search of birth records may reveal other children born to
a couple.
So while it may be more common these days for adopted children to have a delayed or amended birth registration instead of a regular birth certificate, once you get back to births before the 20th century in the USA, most people don't have a state-issued birth certificate unless they specifically asked the state to issue one for them. Their birth record will be at the local level -- if they live in New England, it might be a town record (those can go back quite far), or a church record, or their might only be a record in the family Bible.
For that time period, I wouldn't automatically assume that someone was adopted simply because they had a delayed or amended certificate. An amended certificate might be issued to someone if their parents didn't decide on a name until after the initial birth record was made. Some cultures don't name a child until what we might consider a long time after birth (as much as two years after).
One caution I would make is to set aside the term "real parents" since it doesn't convey any useful information. For research purposes, I encourage you to record exactly what the historical records say. When you refer to someone's parents and must make a distinction, "birth parents" or "bio-parents" vs. "adoptive parents" or "foster parents" (for those cases where you know that is what happened) is more neutral than saying "real parents". Describing what a record actually says, and being careful to note your own assumptions when you write them, will help keep things clear.
From the obituary in the newspaper, you have the names and residences of two sisters. Studying the siblings as a group may give you clues that you wouldn't find by searching for A. T. Russell by himself. Determining when his sisters were married may help narrow down which census records might show the family as a family group. Narrowing down the time frame for when A. T. was taken in by his adoptive family will help set the bounds for the death of his biological parents.
For every piece of information you find, add it to the family timeline, and consider what other records you might be able to find that would confirm that information. Examine each new find to see what might be a clue to something else. For instance, the newspaper gives his church affiliation. It isn't guaranteed that a person worships at the same denomination that his parents did, but it gives you a place to start from.
Here's a list of the records that have been mentioned in the question, bgwiehle's comment, and Rusty's answer, in reverse chronological order. Working backwards from the person's death in small steps makes it easier to follow the trail.
- "A. T. Russell Rites Tuesday" published in the Times Recorder (Zanesville, Ohio) on Tuesday, August 25th, 1942, page 8, column 4, accessed on Ancestry.com 12 Sep 2014.
- Ohio Death Certificate from the collection Ohio, Deaths, 1908-1953 on FamilySearch.org
- WWII Draft Registration Card also found on FamilySearch.org
- 1940 Census from Zanesville Ohio for Allen T Russell on FamilySearch.org (showing a Tennessee birthplace, and clues that Allie previously lived in Wisconsin and Illinois)
- 1927 Death Certificate for Ida Morgan Russell on FamilySearch.org has Elizabeth Cockerham as the informant
- 1900 Census from Magisterial District 6, Allensville, Hadensville (south part) Guthrie city, Todd, Kentucky, United States, on FamilySearch.org Ale T Russell in the household of Thomas F (sic) Russell, wife Ida, children Lizzie, Eula, Ale, Irma, Annie.
And the birth records (found on Ancestry.com, linked to the tree posted in Rusty's answer):

Note that Ida is the mother of 7 children, five living, and there are five children in this household. Searching for other records about the entire family, including the records of the children who were born and died before 1900 might help resolve your question about whether Ida Clifton Morgan Russell is actually Allie's birth mother. No matter how well-documented this family might be, there are still questions to be answered -- the Ancestry tree that Rusty links to only shows six children, so if the 1900 Census' child count is accurate, the author is missing one of the children who died before 1900.
(From my original answer)
Here are some general guides for researchers working in North Carolina:
The State Library of North Carolina's Genealogy Research page has a lot of resources that can help you get started. See the downloadble PDFs for:
Other Guides and research portals:
- The Library of Congress Resources for Local History and Genealogy by State: North Carolina
- The FamilySearch Wiki's article North Carolina
- BYU's Research Outline (somewhat dated, but still useful) for North Carolina
There are similar pages and guides for Tennessee.