This seems like older English. It is not common now, but I believe "linen" used to be used almost as a synonym for "clothing".. e.g. he took off his linens. In this case I believe that "linen" is referring to the shirt, which could be made of linen. So "change your linen" implies "take off your shirt" (and thus get rid of your illness). I hope this helps. The dictionary defines one meaning of "linen" as clothing or household articles made of linen cloth or similar Mitch

I believe "changing the linens" refers to making the beds, changing the sheets. It's a bit old fashioned - I haven't heard it in decades it seems, but once upon a time people referred to the sheets and pillowcases as "linens". When people were sick, it was traditional to change their linens, on the theory that the sheets might have germs. I'm not sure about the "shirt", which would not be normally considered a "linen" unless it was actually made of linen, which it usually is not.

Some people refer to bedsheets as linens (linen is a kind of cloth), but in the context you gave, it sounds like the author is referring to the shirt.