Alternator, battery? Who knows...

Alright guys, let me start off my listing car stats...

1999 Mercury Cougar V6 2.5L (automatic)

I bought this car about 6 months ago, and have had the issue since I bought it. Every time my car would shift down, the headlights and dash lights would dim. Just this past week my alternator went. We finally finished fixing it today, and took it out on a test drive.

Everything seemed fine, actually better than before. However when I first turned my lights on tonight, the dash lights began to flicker rapidly. Then it would stop, and then when my car went to idle, it would start again. I'm going to drive it later when it's darker to see if the headlights were doing it as well, so I will edit the post when I get back from that with my findings.

Anywho, as anyone who changed an alternator on this car knows, it was a b!tch and a half to do. But it's brand new, and we tested it before even purchasing it so we know it's good. I guess what advice I'm seeking is why would the dash lights be flickering when I've just changed the alternator? We didn't find any major fuses blown, so I'm not sure if that could be an issue. *sigh* I love my car but I can already tell the transmission is going on it.

This is just a side note but every time my car shifts you can hear a click (very quiet) coming from the engine/transmission.

SOOO yeah. Advice, help, anything is welcome.

EDIT: So I came back from the drive. The lights flickering also include my headlights, but only when my car is idling. I also got thinking about this but after the alternator was in and everything was ready to go, we drove about 31 miles (to a store and back) and I was wondering if that is enough time for the alternator to have charged my battery up, or if it could still use some more driving?

I questioned this because as I was pulling out onto the road just now, it sputtered like the battery was dieing, and kind of jerked a little (I've had this happen before my alternator was changed.)