I was in the same situation just down the road from you in O'leno State Park.
We were camped under huge deciduois trees and MANY snapped trunks still standing. When we heard of the storms coming, my reaction was the same. "I am not sitting here under all of these trees.
I went to find some clearing we could go to and there was none in the whole park. Even the parking lot was lined by huge trees.
We actually cancelled our remaining reservations and moved to a different park 30 miles away.
We still spent time WITH OUR PETS in the shelter until the storm passed. I just felt much safer not having to worry the whole time waiting for the storm to hit.
I will add that I had recently purchased a little portable weather radio that was blaring out warnings on a regular basis. I was very glad I had it.
NOTE: We were the ONLY campers in the shelter/restrooms. I am not sure what goes on in other peoples heads but sitting in a camper in high winds, hail, T Storms, Tornado Watches/Warnings, is not for me. - to each his own.