Car starts, then slowly dies while im driving

I have a 98 ford taurus. It needed a new battery so
it was sitting for about a month. We got a new
battery installed and first day it drove fine, next
morning it drove fine, then on my way home from
work it slowly died while i was driving. I didnt
notice till i tried the turn signal and it didnt come
on, and then i noticed all dash lights were off and i
couldnt accelerate, so i pulled off the road. I turn it
off and take the key out, then try to start it again as
normal and it just makes the dead battery clicking
noise. I pop the hood, check the cables, check
everything else, everything seems fine. After sitting
there for about 10 mins i try to start it again and it
starts right up. I drive for about a mile and it dies
again. I sit there for a while and yet again it starts
right up. I manage to pull into my parking space
when it dies again. I noticed that the battery light
and the parking brake light have stayed on since
we installed the new battery (no, parking brake is
not engaged) and i noticed that right before it dies,
the abs light comes on and the air bag light starts
flashing. This morning i used a baking soda
solution to remove the corrosion from the positive
connector, as there was quite a bit, and it didnt fix
it. I have considered it may be the alternator, my
boyfriend agrees, but i also found other
possibilities while doing a quick google search. I
hope i have provided enough info to narrow it
down.