Only alternator but test both
When engine is running the alternator provides all the power/voltage for the vehicle
Engine running voltage should be minimum 13.5volts, just after start up it can go up to 14.8volts but only for a few minutes
A battery's brand new voltage is 12.8-13.0volts, so with engine running(13.5v) voltage flows TO the battery to keep it charged, no voltage should come FROM the battery, battery only used to start the engine
As said new battery is 12.8v
3 year old battery 12.5v
5/6 year old battery 12.3v<<<time to shop for battery sales
12.2v or less is a drained or failed battery, change or charge
You should test battery voltage AFTER battery has been sitting unused or charged for at least 4 hours, the long the better to get a true voltage reading
Test the battery first, to get base voltage to test alternator wiring
Lets say battery voltage is 12.4v
Use volt meter grounded to alternator metal case for all tests
Key off
Test voltage on B+, the stud nut terminal on the back of alternator, should read battery voltage, 12.4v
If not a fusible link is blown
Unplug 3 wire connector on alternator
Test yellow wire, should read 12.4v
if not a fusible link or fuse is blown
Test light green wire
Should be 0v
Turn key on
Retest green wire, should be 12.x volts, can be a little less that 12.4
If not wiring from cab is bad, this is the Battery Light circuit, and the ON/OFF switch for alternator
If all wires test as OK then plug wires back in and start engine
Test battery voltage, should be above 14volts, that should last for 5-10min and then drop below 14.0v but no lower than 13.5volts
An alternator has 3 Fields in its metal case, each one generates 1/3 of the voltage
Its not uncommon for 1 Field to fail as alternator gets older, you would notice dimming head lights at idle but system would work fine with 2 fields when driving, RPMs above idle levels
If 2 fields fail then there is a problem, similar to what you describe
A failing alternator can cause the battery to fail early because its either over charging it, or draining it
A failing battery can cause alternator failure because alternator has to run at full output all the time