Easy fix for less than a pro would've charged
Il Mio Problema
Battery worn out, and OS said to "service soon".
La Mia Riparazione
Ordered replacement from iFixit and installed it with my own tools. A couple screwdrivers, a spudger, and a few minutes time, and I have a new battery! I did have an odd instance of the computer suddenly sleeping (very shortly after typing and clicking), but I reset the SMC, and it's been working fine since then. OS now says Battery "Condition: Normal." So far, so good!
Il Mio Consiglio
If you have trouble with the computer randomly sleeping, etc, you might try resetting the SMC: https://support.apple.com/en-us/HT201295