Either can work. I have done a good bit of touring with ACA maps, some with a dedicated handheld GPS, and some with my phone supplementing paper maps. For the TA I'd use the paper maps if doing it again. I would have the cell phone along so a quick google search would be possible, but I'd probably keep it turned off when not in use to conserve battery.
The TA route is straight forward enough that you really don't need to be constantly watching for the next turn. One of my companions seemed to be able to remember the whole days route pretty reliably most days. I just kept the current map in my map pouch and followed it.
I loved using GPS on a tour where where I was riding dirt roads and trails most of the time and there was little to no signage a lot of the time. I did find keeping batteries charged to be a hassle.
Also I'd miss all the info on the back of the ACA maps.
As far as the bulk and weight of the paper maps... I pack pretty crazy light (my most recent tours were in the 9-15 pound base gear weight) and still felt the maps were worth carrying on road tours. I typically do mail them home as I finish with them and if you will be having any stuff sent from home along the way you could plan on having some of the maps sent to you. If worried about the weight or bulk, you could consider taking maps enough to get you over the mountains at the end you start from and have the rest sent to you via general delivery when you hit the plains.
I am not sure where the ACA is on this, but a dedicated phone app that would allow all the features of the paper maps along with the advantages of the GPS would be the best of both worlds if you can manage the battery issues well enough. I used an app like that on when hiking the John Muir Trail. I carried a couple extra phone batteries and a small power wallet and left the cellular part of the phone turned off.