Next, I called GEICO to tell them about this quote and ask if they can offer any discounts to close the gap. I found out that the discounts they offer vary by state, so apparently, there was one discount I got in NJ that I'm not getting in DE. Weird. While going through the list of available discounts, I was tempted to ask if they had an AAA affiliate discount. :) Anyway, the customer rep couldn't offer me anything other than a discount for taking a defensive driver course, which I wouldn't have to take to get AAA's lower auto rate. So it looks like I'll have to switch.
Another option is to get auto insurance from AAA and homeowner's from GEICO since the latter is cheaper there. But that would be messier and probably won't save much because then I'd lose the multiline discount.