It sounds like the Senators were the ones with the grounds.
Although I can't find the exact year, I read that the Walgreens "W" came out in the 50's, but suing a struggling American League team in Washington wasn't on their radar. I do note that the Nats' "curly W" is angled upward towards the right. In 2005, a Nationals PR guy said the logo was tilted up towards a new beginning. (Others would wag that it's the stern rising out of the water.) I would not be surprised if part of the reason why this was done was to ensure that there would be no trademark infringement lawsuit by Walgreens.