Hi, Katie. Just a quick comment rather than a full answer. Thomas Morgan is way too far back in time for you to carry any identifiable DNA of his, even if you are a direct descendant. To align ourselves with ancestors, since we can't test their actual DNA, we're limited to finding matches with cousins contemporary, more or less, to our own generation. By the time we get to 6th cousins--so shared 5g-grandparents--the expected average sharing is only 0.01%...so effectively zero. The amount of sharing decreases fourfold with each cousin distance (2nd to 3rd to 4th) because the DNA recombination and mixing is happening with each birth event. If the generations average 30 years, 5g-grandparents lived roughly 210 or so years ago. It might be possible, with a lot of work, diligence, and multiple cousins to test, to get back to the late 1700s. But that's functionally as far back as autosomal DNA can be used for genealogical purposes.
Best of luck with the search!
