My incoming calls only allow 4 rings before going to voicemail

You will need to change the number of rings to forward to VM in the AT&T Portal.  See below last section "call forwarding":

The portal is intuitive and often self-explanatory. Each grey arrow indicates an action needed to customize the feature. For example, under Call Forwarding in the first image, there is a grey arrow next to No Answer Call Forwarding. On the same line, OFF indicates the feature is turned off. To turn the feature on, click the grey arrow to take further action taken as seen in the second image. When making a change, always remember to Save your changes.

Check your call history. Block unwanted calls. Change the number of times your phone rings before rolling to voicemail. Make U-verse Voice work for you, anytime, day or night.

The answer supplied looks all well and good, but I remain in ATT hades with "How-To's" that describe using "nn#" and "*nn"

I cannot  find any path that brings me to the easy-to-use screen you display.

You need to open your U-verse account and go to the drop down on the upper left...I want to. Select U-verse voice and manage features. This will bring you the screen in BeeBeeSA post. Open the tab for voicemail and you can control your VM features.