I may have corresponded with you previously, but I don't have mention in my notes, if so. Please take a look at my message #9702 in this forum to see if my inferences have meaning to you.
My Alexanders are not closely related to yours; however, my wife descends from Nancy (Alexander) Dysart of Burke (later McDowell) County, NC, and I believe she may have been Joseph Lowery Alexander's daughter. She and her husband lived by Joseph Alexander (appropriate age for her father) in 1820, and they named their oldest son Joseph Lowery Dysart. Tate was also used as a Dysart middle name in the middle 1800s.