Stargazer's saucer is five decks thick in the same places the Constitution's is only one, so that's actually ALOT of extra habitable volume even before we consider that it's also a bit WIDER than the Constitution class. Put another way: if the widest deck of the Constitution's saucer were the "main deck" where crew quarters were located, Constellation has FIVE of those decks, plus the same additional smaller decks as on the Constitution, plus a largish dome on the underside that probably houses some kind of stellar cartography complex. The enlarged saucer section more than makes up for the lack of a secondary hull as well, especially where it extends behind the saucer and into the pylon mounts.
According to the old "
Starship Volumetrics" web page (whose figures are questionable in a number of ways, but still) Constellation should have about three times the overall volume of the Constitution. Make of that what you will.