I definitely think you could be able to obtain an ADR. The best advice I have for you is to keep looking and check back a day of two before you leave, many ADRs open up 24 hours prior to the date of travel. I would also consider looking into touringplans.com's reservation finder. You input your desired dining time and location and if it opens up, touringplans will notify you by email or text.
As for if there is a difference between calling or booking online, the answer -- not really. If you call they can check availability for a span of many days at a time while if you book online you can only look at one dining period (ie breakfast or lunch or dinner). Other than that there are no differences.