But if the battery hasn't "died" completely yet, then wouldn't the clock
losing time be a symptom of that?
No. Either the battery will suplly current or it won't. There's no
middle ground here.
On a side note, if you have a computer produced within the last few
years, if you turn of your computer with it's power switch, or use
Start/Shutdown, you are NOT interrupting power to the MB. The PS
still trickle power to the system. This lengthens battery life,
since no load is being placed on it.
The only way you'll ever use your CMOS battery is to interrupt power
to the computer by pulling the cord from the case, hitting the PS
on/off switch (in the back of the machine), or switching off the power
supplier the computer is plugged into - something like a power strip.