So my dog loves being in water and she will end up wet regardless of rain.
My first defense against rain is her rain jacket. I also carry lightload towels, and they are extremely small. I use them when we get in the tent and then again before bed if needed. Finally, I bring a lightweight fleece, like a cheap one from Walmart, for her to sleep on. That also helps dry her and acts as a barrier for wet fur on my quilt.
Edit - post bleeped my link so I deleted it