2006 Honda Civic car battery dies every 2-3 days

About two weeks ago, my car wouldn't
start. We thought it was a battery issue so
we got a new one. A few days later, it
wouldn't start again. Every 2-3 days it dies,
but I can get it started again with a jump. I
took it into the shop and they couldn't find
the problem. They said the battery was fine
and couldn't find what was draining its
energy. I just went to a Honda dealership
and they said that it was likely the AC
compressor. My AC has been out for over
a year (I got it fixed previously but it broke
again so I just accepted that I wasn't
gonna pay for AC). Anyways, they are
asking $1300 for parts and then another
$700-800 for service. Ahh! Even if I do get
that all fixed, it might not solve the
problem. My question is--do I need to
catch the whole AC system replaced, or is
there a cheaper way? Could I get a new
relay? Or somehow switch something off
with the AC system? Any advice is
appreciated!!