If you have a new battery installed and as you run the engine, the battery is draining, then the alternator is not charging the battery (so the engine is running off of the battery).
I can think of two possible reasons why the alternator is not charging the battery:
1) the alternator itself is not working properly
2) the belt that drives the alternator is 'slipping'
The Factory Service Manual has instructions for testing the alternator. Or you can try calling a local Aftermarket Autoparts Store and see if they do free alternator/battery testing.
The FSM also shows how to check and adjust the tension on the belt that drives the alternator.
The only relationship that I can think of between the symptoms above that you attribute to vac related and alternator related, is that the same belt drives the Air Pump that drives the Alternator.
If a component, in this case the Air Pump, is seizing up or has seized up (it has happened to me), then when it seizes, the crank pulley will still try to pull the belt around that seized pulley on the Air Pump and the belt will get hot. If the belt is really tight, the seized pulley on the Air Pump can prevent the crank from rotating, so the engine will die...