Nissan Altima 2009 - Battery or something else?

I drove 15 minutes with no issue.  Was in the store for 20 minutes.  When I
came out, my car would not start.  Jumped it and was able to drive for 10
minutes (not long enough to charge it).  So, I was not surprised when I
needed to jump it again.  I was able to drive 20 minutes home with no
problem.  The next day, it would not start.  We jumped it and planned to drive
it for a good 40 minutes.  But, after we back down the driveway, we couldn't
get it to move.  it sounded like we were giving it gas and it was revving, but
but no movement.  After we let it run for a few minutes, we decided to try
again and still no go.  I turned the engine off.  Just for curiosity, I pushed the
start button and the car turned over fine and was responding normally to
acceleration.  We drove it for 40 minutes.  We let it sit for 4 hours and tried
again and it was fine.  The next day, it's dead again.  This time, it would not
sense my key fob, the power locks would  not manually lock or respond to
the key fob.   Didn't have time to try another jump this morning.  The battery
old (could be the original).  I did observe corrosion.  I'm wondering what my
next step should be?  Does this sound like I need a new battery?  or does it
sound like there is a parasitic drain?  Of note, my AC has been not working
all season and I was told I need a new compressor.  I have had the AC/Heat
in the off position, so it  hasn't been running.  Just throwing that out there in
case it's related.  THANKS!!!