Is my battery too weak???

I took the battery (high quality Interstate) out of my 95 Olds Eighty Eight and put it in my 99 Olds Eighty Eight. It works perfectly fine, but it seems like the battery may be too weak for the car. Example: when I turn on my hazard lights and turn on the interior lights at the same time, you can see the interior lights slightly blinking along with the hazards. Also, sometimes, it sounds like the car will heave/surge power, everything turns SLIGHTLY dim for a second, and then goes back to normal.

The funny thing is that when I'm shopping for a battery, both cars use the exact same batteries. Is my battery weak, or is it something else (alternator maybe)? What do I need to look for in a battery when purchasing?

Thanks!

Um, have the alternator tested for loaded output, also test the battery, make sure all the connections are clean and tight, basic stuff here.