Typically, reports are that you can make your online ADRs starting at 6am Eastern Time, and via phone at 7am Eastern Time.
I would recommend prioritizing which ones are most important to get for specific days and times then do them in order of priority. If you have to do Credit Card Guarantee for any of the restaurants (usually more popular ones and character meals) it's faster to do on the phone vs online, but not impossible online. The reason being is that online you have to enter in your CC and billing address for each reservation but on the phone they will do it one time at the end. Saves your fingers some typing.
Regardless, have a plan, stick with the plan and I'll bet you'll be pleasantly surprised. I think on my last trip (June), we had meals for just about every day of the 10 day trip scheduled and I got every single one of them within 15 minutes of when I wanted to schedule them. Mostly did them online except for Fantasmic Dining because for some reason that wasn't available online for booking but was via phone.