Dallas is an Eastern city in only the loosest sense of the term - that defining the East as the Eastern and Central Time Zones, which would put part of Texas in the Western US.
Back when the Rams were in St. Louis, I would have put the Cowboys in the NFC West with Arizona, San Francisco, and Seattle - Cowboys being a distinctively Western nickname. It would have made the rivalry with the 49ers all the more intense.
Are Dallas teams in Eastern conferences or divisions in any other major sports league? No. So therefore, the Cowboys shouldn't be in the NFC East either. I do advocate a four-league realignment of MLB which will stop the Texas Rangers from having to play 9 series on the West Coast every year as they do now (and for that matter, the Houston Astros).
Dallas has more in common with Atlanta and New Orleans than NYC, Philly, or DC. NC teams share divisions with DC teams in the NBA and NHL as well. Plus, former Panthers coach Ron Rivera now coaches Washington. Rivera would relish coaching against his former team twice a year.