I had something similar happen using the Accuweather widget.
Occasionally, my location within the widget would display incorrectly though my physical location had not changed. I discovered (eventually!) that either I, or the widget itself, had inadvertently added my location with the incorrect name.
You have to get to the City List display and remove any undesired location names.
To check if this is what is happening in your case:
1) Tap the Accuweather Widget background to get to the Detail View
2) Tap the List View icon (currently at the top and immediately left of the '+') to see City List
Do you see any incorrect or undesired City Names listed? If so, you have the same problem I did!
3) Select an undesired City Name and then tap the Trash Can icon to remove it from your list.
Remarkably, I was able to resolve my Location/City Name down to the neighborhood level using this technique. Once you've eliminated any names you don't want, use the '+" to search for any names you do want.
Be careful, overlapping City Names/Neighborhood Names will cause random switching and prove to be as frustrating as OP's initial problem (IMHO!)
Good Luck!