Sadly, this is one of those cases where you may never know where Lizabeth came from or who her family was. The Civil War caused immense devastation across the South. Families ended up homeless and wandered looking for work and food. When the adults died the children wandered by themselves. There are countless stories like those of your Lizabeth. My own great-grandmother has a similar story.

You could try a DNA test. The test you would need to take is the Mitochrondrial (mtDNA) test. The largest database for DNA tests is currently held by FamilyTree DNA, which was established specifically for DNA testing. They have special price offers in the summer and winter.

This is a good summary of DNA testing:

http://www.jgsws.org/Handouts/2-11-13_IveHadMyDNATested_NowW...

One confusing part of your post is that Lizabeth was 12 years old in 1873, but her tombstone states she was born in 1826. What was the approximate birth year from the censuses?