Clean your grounds.
Might also have corroding cables or battery terminals.
I'd check the voltage drop at power points and use different grounds to figure that out.
Especially true if it does this at operating rpm, not just at idle. You have no reason to suspect the alternator yet. This screams bad ground or cable to me.
You can also use a single jumper cable, if you don't have a meter. Connect from the
-Bat to the engine block, try it. Connect again to the chassis (bare metal), try it. Connect again to the frame, try it. Then go to B+ and connect to the alternator + post(big red wire), try it. Then B+ to the solenoid (follow the cable from the battery so you don't crank the starter if you use the wrong post on the solenoid). You're basically just duplicating an existing power or ground path. If it improves with a jumper, that run of cable, ground, etc. is bad.
Check your alternator belt too.