Which "Senators" do you mean Snives? There were two, after all …
Senators Mach One - relocated to the Twin Cities, circa 1960. Harmon Killebrew, Bobby Allison, etc.
Senators Mach Two - started up the following year in expansion, managed later on by Ted Williams. Frank "The Capital Punisher" Howard, Mike Epstein (who had an even more colorful nickname I won't use here, lest some folks get triggered), and Don Mincher were probably some of their big boppers. Howard had an amazing run in DC over 6-7 seasons, but when they eventually moved to Texas (Rangers) in the early '70's it pretty much marked the end of his career.
Not sure which one was the one you referenced? Do realize the difference may age you by 10 years.
IIRC the Sens' last game before moving to Texas resulted in a forfeit when the crowd prematurely invaded the diamond. Having been in the Big Owe for Les Expos' final game 30-something years later, I can tell you there were moments during that game when I wouldn't have been surprised to see the same thing - 30,000-plus angry Montrealers, and as luck would have it, the last visitors were the Marlins - then defending WS champs, and owned by the father/stepson duo that pulled the plug on an approved new Montreal stadium a few years earlier, before they dumped the team on MLB in the same transaction that sent John Henry and his crew from Florida to Boston. There were long stoppages mid-game, and it's the closest I've ever been to a possible real-live riot.
Hated to see it happen to the Chargers a couple of years ago, and yeah, if Tampa Bay ever leaves for Montreal (or anywhere FWIW), I'd be a hypocrite if I didn't feel badly for the fans they leave behind. Folks get attached to teams. It's gut-wrenching, if you ever have the misfortune to experience it. There are WAY more important things in life, but it still takes a piece of you away.
So yeah, DC was without a team for awhile, but I suspect the one you remember left for Texas, not MSP.
Maybe the you-know-who's will prove the 3rd time's the charm for DC baseball.
I still hate 'em though.

Can't help it. I thought I was over it, found out this week I wasn't.