Hi, Lisa. I assume you're looking at autosomal DNA. In which case, another contemporary descendant of your 4g-grandfather William Hawkins would make the two of you 5th cousins. And at that number of degree-of-relationship separations--12 birth events between you and your cousin--autosomal DNA really can't be used as evidence one way or the other.
Fifth cousins would share a mathematical, theoretical average of 0.0488% of their DNA. From a practical perspective, it's likely that the amount shared is too small to even be detectable. A nifty study by Brenna Henn, et al., showed that there is less than a 15% chance that 5th cousins would share any detectable autosomal DNA at all.
Some folks feel that with extensive triangulation autosomal DNA can be used as evidence to the 5th-cousin level. But a lot of knowledgeable people disagree with that. At the very least, though, a non-match with a 5th cousin really means nothing.