I did take a quick look at this, I did not find that much. For some reason I can't bring up any Washington County GA records in 1890 on ancestry.com, I think the only 'common' genealogy documents that can help you out here are 1890 and 1900 census.

I can't find her in 1900 census, I can also not find anything suggesting she (and also her daughter Izora) had a SS# by looking in the social security death index. I also can not find anything about where the Buck and Hall surnames fit in. And also not anything about the younger brother Ralph (but I only looked for him in Washington County). If Rose really had a younger brother Ralph, it could mean that her entire family is missing in 1900 which would be unfortunate.

In 1930, there is a cousin Susie Veal living with Rose Peeler. By following her back through the census to 1900, there is a nephew William Walker living with her. So there seems to be some kind relationship between your gr grandmother and the Walker family just as you had heard. Also in 1900, Susie Veal is living one household from Richard and Anna B Buck. Maybe some relationship there too? Going back even further, I think Susie Veal in 1930 is Susan Dixon in 1870, living in the household of Augustus and Betsey Dixon in 1870. If true, then either Augustus or Betsey could be a sibling of either of Rose's parents. Maybe Augustus since he carried the name Dixon.

I can also see the not-so-common names Amos and Isadora Dixon in 1870, maybe Rose's children were named after them?

Since I don't have any real clues here, I would look at more vague possibilities.

Walter Peeler is living in city of Deepstep in Washington County in 1900 (so is Susan Veal). Since Walter and Rose married in 1902 and they might have known each other for a year before that, and also that Deepstep seems to quite rural, I would assume that Rose was living in Deepstep as well in 1900, but maybe not listed.

There is a Dixon family in 1900 living a couple of pages from Walter Peeler: Clay Dixon and wife Sarah, they have two daughters Matilda or Fannie born in the mid 1880s. Could one of them be your gr grandmother Rose for some reason listed with a different name? Did rose have a nickname that reminds you of Matilda or Fannie? Since the 1910 census says that it was Rose's first marriage, I do believe that Dixon is the surname you should be focusing on.

So here is what I would do:
1) Try to find an obituary for Susie Veal, if you're lucky Rose Peeler is mentioned and if you're really lucky some of Rosies siblings (if she had any) could also be mentioned.
2) Try to confirm that Susan Veal in 1930 is Susan Dixon in 1870.
3) Try to find any marriages for Matilda or Fannie Dixon after 1900 just to rule out that anyone of them are really Rose.
4) Try to find the local church in Deepstep where the Dixon's and the Peeler's might have attended. Maybe anyone elderly in your family knows?
5) Try to verify if that church has kept their membership records from that time (possible, but not common) and if so, try to find any Dixon families listed as members.
6) Try to find some old decendants of the Walker, Veal or Dixon families still living in Washington County, GA.

Maybe you can tell more about what you have found/not found in 1890 since you are fortunate that the census still exists.

Nothing of these are any hard clues , but hopefully it could give you some kind of starting point. I think that if you find out who Rose's siblings were, you will find what you're looking for in 1900.