Update: I went to a yard sale today and they had one of those root waterers that has a 4' long metal spike you can drive into the ground to water a tree. Got it for $1.
The tater towers were definitely dry, despite the last 3 days me soaking them with the hose over and over. I put that thing down in there and ran it for about 10 minutes per tower until I saw water coming out the bottom.
Now I understand why some of the howto videos I watched they put a PVC pipe down the center. Now I get it. Do that, and you can drill holes every few inches, fill up the pipe and it slow waters your tower. I'll be adding that next year.
I think the issue was no matter how much water I poured on the top, it wasn't getting down into the center of the mass of dirt.