Oh dear, well it certainly sounds like the battery
You mention that you've tried another battery - was this a new one or could it be that you inadvertently have 2 failed batteries.
Once the vehicle is started and has run for a little while, will it turn off and immediately start again?
You also changed the alternator prior to having 'this' problem. I wonder if that is a coincidence..... Is it worth double checking that there isn't some kind of short circuit around this area??
Just a few thoughts.....