Just be ready the day you can make ADRs in the morning. One thing with CP reservations is that they may not be available to reserve when your 60 day window starts. Last year our arrival day was the last day Brendan Fraser was doing the show so we wanted him (sorry Joey McIntyre) but they hadn't dropped CP reservations into the system at our 60 day window. The day they dropped the CP reservations my girlfriend and I set alarms and both tried to make Le Cellier reservations. We were able to get it at roughly the time we wanted, I will say that I think Le Cellier reservations dropped like 10-15 minutes after the other restaurants for some reason. I just remember right at 6am none were available and I booked Coral Reef instead and a little later Le Cellier dropped.
For phone or laptop, I'd just use whichever is more comfortable. You can have more than 1 person searching, at least we do but we were laying next to each other in bed so we could talk to each other to plan easier.
I agree with Marthasor, depending on the narrator will determine the ease of the reservation. Brendan Fraser was a mad house and we glad to have the reservation, later in the week we could've just walked in. We chose Le Cellier because, while yes very expensive, we figured that we may not eat there without the package and by looking at what we wanted to eat it was only like a $15 premium w the package and there was even a higher premium at Coral Reef.