First of all, don't just set the CRS of a layer.
If you want to change the CRS of a layer, you should re-project it. The actual steps to reproject a layer depend on your GIS software. However: before re-projecting a layer, that layer should already be in its native CRS. See below.
Usually when you load a layer into a GIS software, the layer comes with the CRS already set. If it doesn't, you have to figure out what the native CRS of that data is, and manually set it to that CRS.
The Google satellite layer is probably in EPSG:3857 (Web Pseudomercator). That's the normal CRS for Google maps layers. Open the layer properties and change the CRS to EPSG:3857.
If you changed the CRS of the layer from the city website, remove it from your project and add it back in. If you're lucky, it will automatically load in its native CRS. If it doesn't you'll have to figure out what its native CRS is and set it manually. Or you can try re-downloading it and see if it loads with the correct CRS (just in case you accidentally changed the default CRS for that file).
The city website should tell you what CRS their layers are in. I don't see that information on the page you linked, so you may need to dig around on their website, and possibly reach out to them through the contact information.
While waiting for them to respond, try putting the layer in every available Washington state plane CRS. Then try the UTM projections for the correct UTM zone. Then try as many more different options as you have the patience for. (Stop experimenting when you find the one that makes the layer line up correctly with the basemap.)