Mona was driving toward the hospital, beside herself with worry for her husband.
Sounds totally natural and correct. Here is the definition from Merriam-Webster in case it helps.
Mona was driving toward the hospital, out of herself with worry for her husband.
This one I haven't heard before. Maybe you meant "out of her mind with worry"? I searched for this to see if it is common or if I could find a definition, no luck. The most similar phrase I could find means something entirely different (to talk oneself out of something OR to take you out of yourself).
I took a look at Google NGrams and it seems to confirm that "beside herself with" is more typical than "out of herself with".