You may be on the right track when you say it stays charged if disconnected.
Will it start the car if disconnected for a longer period, such as over a weekend? If so, the battery capacity is probably OK.
If the battery itself is OK, you must do some detective work to find out what is drawing current when it is shut off. Try searching on other battery-related threads here for some tips on detective work. They involve looking for current use when there isn't suppost to be any, such as a trunk light that doesn't go off even with the deck lid shut.
I wouldn't suspect the alternator, unless it won't put charge back into the battery.