You should be aware that widows and veterans of Federal regiments from Tennessee would not/could not apply for a state pension. The only applicants to the State of Tennessee would have Confederate veterans and their widows. Federal veterans applied to the U.S. government.
Many applications for veterans and widows of veterans of Dibrell's/Gore's Tennessee Cavalry will call this regiment the 8th Tennessee rather than the 13th. That's what it was called during the war.
You may be interested in the regimental history here which explains some of these issues: