My cousin Derek agreed to have his DNA tested and we are a match. Specifically, we share 18.6 centimorgan's on chromosomes 15 and 16 so this must be the DAWES heritage.
Family Tree DNA estimates that we are 4th cousins while GEDmatch puts it at 4.8 generations to a Most Recent Common Ancestor. From our paper records, I believe that our common ancestor is Thomas DAWS b. 1773 in Westham, Sussex, therefore, we are 3rd cousins once removed.
As you can see from the above chart, we have one extra generation between us hence the "once removed." According to the relationship charts available at http://www.thegeneticgenealogist.com we could have anywhere from 0 to 332.26 cM so that fits our 18.6 result.
The next step will be to try and find others on GEDmatch who match us in the same area. Stay tuned...