Battery Saver Active Car loose power but then comes back

What is your question?

My 2008 Impala SS will show the Battery saver active message and after a while will almost completely die. All lights and gouges will turn off and the car will stay driving until I slow down to turn. Then it will stutter and act like it wants to go but won't. If I floor the gas pedal it'll stutter for a minute but then it'll take off and all power comes back like nothing happened. I have a new alternator I can put in but should I try a new battery first? If so, any cheaper alternatives for a new battery? All I find is 199+ for a new battery online and part stores.

What makes this problem better or worse?

Not charging battery for a week and the cold

How long has your 2008 Chevrolet Impala had this problem?

A few weeks