Spider mites certainly like rosemary, Louise.
Finding them, is difficult. They are tiny, tiny insects. Yes, they make webbing but that matches their size. Finding them before they have caused lots of damage takes careful looking, that is for sure.
It is almost like staring at a little spot and waiting to see some movement. It sounds like you have been patient and done that.
The first thing I'd do is take a hose and spray the tiny critters off - as best as I could. If some rosemary leaves fly off, that's fine. They were obviously not very well attached or doing the plant any good.
I would use an insecticidal soap spray on a plant that I much value. I will use dish soap on sunflowers and that sort of thing but dish soap is for cleaning dishes and you have to be concerned about what it might do to the plant. I use Palmolive Green - mostly, just because I've used it so many times before. There may be better choices.
Here is some information on using soap on garden plants [url=https://www.ext.colostate.edu/pubs/insect/05547.html]from Colorado State University (link).[/url] Notice their cautions about damage. I will spray in the late afternoon and like to return the next morning to rinse the plants.
CSU recommends 2 to 3 percent dilution with water and Table 1 gives you the amounts. I use 3 Tablespoons per gallon.
What I do is:
Wash the plants - strongly - with water. Wait for them to dry then spray with the soap mix. Later, rinse the plants.
I hope that this is of some help.
We are each other's harvest; we are each other's business; we are each other's magnitude and bond. ~ Gwendolyn Brooks