Does signal strength fall off as the shifter batteries lose charge?

No. Shifting performance is unchanged regardless of remaining battery charge. The components do provide indicator lights for battery life. After each shift, a light on each component turns on.

Shifters:

Green LED: 6-24 months

Red LED: 1-6 months

Flashing red LED: Up to or less than one month

 

See the SRAM RED eTap System Overview video for a visual of the indicator LEDs.