It is worth shopping around.
I had AAA in California, but when I moved to Las Vegas, local AAA agency quadrupled my payments.
So I moved to Geico.
Now from time to time I run comparison and most of other offer who "saved $500 over Geico on average" give me quote twice as high as my current payments.
Beware about those internet quotes. Each time I run some, my email is getting 30 junk mails daily in main box and probably a 100 in junk mail. Not the insurance quotes, but the "get hard" and similar.
That goes for about 2 weeks before quiets.