The "seachbox" showing or not really has nothing to do with whether Cortana is on or not.
In essence it's cooked into Windows so deeply that you aren't going to turn it off in any kind of complete way. I would suggest that you look into your privacy settings and such to see where your vulnerabilities or security concerns are in reference to this and make appropriate changes there.
There are a great deal of good articles to be found under a general heading of "what should I disable in Windows 10".
Keep in mind that with ANY program you use now, that it may or may not be collecting information or worse. You have to ask yourself whether you trust that turning off certain aspects of Windows (supposedly) is enough, or whether you trust developer "XYZ" making programs that help disable the way Windows is supposed to operate.
Create a new 32-bit DWORD value named AllowCortana and leave its data value as 0.
Since Windows 10 v1803 (Build 17134) (i.e. Windows 10 - 2018 Spring Update) you also need to add the DWORD BingSearchEnabled and CortanaConsent values with a data value of 0 under the mentioned key.
You can also create a registration file to perform the required changes. Just add the following registry code to a file with the name DisableCortana.reg. Once the file has been created then right click on the file name and left click on the Merge option from the context menu. Restart the OS and the changes should take effect.