Brake and battery light on, and battery keeps draining on my Nissan Xterra

I have an 01 Nissan Xterra. My brake and battery light have
been on for some time (used to come on then go off, now it
just stays on). My battery keeps draining as well. I have
bought BRAND NEW batteries, and they last about a week.
My dad checked my alternator by removing the battery
cables while the car was running, and having me turn on
various electrical things. The car still ran, so he says my
alternator is ok. But I'm thinking maybe it is over charging?
Cause I'll buy a new battery, it'll work for a week, go dead,
won't charge, and the store I bought it from will tell me it's
bad. And I've bought then from different places, and bought
different brands, so it's not a bad batch of batteries. My
vehicle is killing them, and I don't know why! The alternator
was bought brand new in 2017.