Not staying with selected theme

I have 2 separate websites that both use the same device theme switcher plugin. One of them works perfect, the other, when using a smartphone and switching to “full-size” view, it switches to full-size view until you click any link on the site and it reverts back to mobile. My settings are the same and method of switching to full-size is the same. I’m stumped.

Has anyone else had this problem and figured-out why it won’t remember which theme to use?

Thanks.