I supported the original and expansion Senators. In baseball's hiatus from DC, I took my kids to Memorial Stadium a handful of times. Never could get into supporting the O's though. Always rooted for their opponent. It's a Baltimore-DC thing.
Your experience does seem typical for a lot of Senators fans. As I noted, though, there were whole generations of fans who grew up with no memories of the Senators.
It might have been different if the DC area had another baseball team in the interim. There's a discussion
here (O's fan forum) on whom Baltimore fans rooted for before 1954. The consensus seems to be the local AAA team rather than any MLB team, including the Senators, although much of that has to do with lack of a TV market, so Baltimoreans didn't have much exposure to the Senators anyway.
Someone in that thread looked at old issues of the
Baltimore Sun to see what baseball coverage there was, and it was mostly the AAA team with MLB coverage relegated to AP box scores. I suspect if you looked at 1980s-1990s issues of the
Washington Post, you'd find rather more dedicated coverage of the Orioles, though still drowned out by coverage of the Redskins and Bullets/Wizards (and maybe Hoyas for a little while).