New alternator makes less power than original one.

theres something wrong. Even with the stock alternator, headlights on, heater on high, and stereo blasting, you should have 'enough' juice. My lights would slightly dim though when the bass hit... but nothing like what you describe.
Double check your wiring, and make sure the contacts for the power wire to the 3G are clean, the wire is good, and the contact to the drivers side electical stuff is good.
Are you sure the 3G is good?

u could have a bad altenator take it to autozone they will check it for free also check the green wire that goes to the sylinoid and make sure it has a good conection i had trouble eith that wire once