No, you didn't need to disconnect before moving.
You just need to tell you Chromecast about the new network now that it can't find the old network anymore. Go through the setup steps again, you'll find your Chromecast is already waiting in setup mode:
https://support.google.com/chromecast/topic/7215248
Two things to keep in mind. Chromecast devices can only remember one network so you'll have to do this every time you change networks, even if you go back to a previous network.
And, if you really do have a 1st Generation Chromecast (the key-shaped one), Google recently announced that they have stopped releasing updates for it. It should still work for a while but eventually you may start having problems with it. Considering the 1st Gen hasn't been sold for more than 7 years, you have certainly gotten your money's worth out of it; congratulations for taking such good care of it. You may want to consider upgrading. The new models have a LOT more features (including a remote and an interface so you don't have to always use your phone to cast), better processing power, stronger WiFi radios, and a design that doesn't put as much strain on your TV's HDMI port.
