1987 Acura Integra Battery keeps dying.

Well here is the problem. Car runs great, though when I turn on the headlights and interior lighting when the car is idling around 1 grand to 800 rpms the cars battery and headlight lights in the dash flicker a little though if I keep it above 1 grand this with not happend. Also when I turn the headlights on I can tell they are draining power and the car will stutter a little though stay running.

So I have checked the battery, its good. Checked all grounds, there good. Checked fusees, there good. Checked alternator, that is good also. Let me know what it might be. Thanks