You always use the DC of the spell, unless said otherwise.
You don't have to recalculate your con bonus, calculate your HD or anything, simply use the spell save DC for any abilities that require a saving throw.
From Transmutation (Polymorph):
Each polymorph spell allows you to assume the form of a creature of a specific type, granting you a number of bonuses to your ability scores and a bonus to your natural armor. In addition, each polymorph spell can grant you a number of other benefits, including movement types, resistances, and senses. If the form you choose grants these benefits, or a greater ability of the same type, you gain the listed benefit. If the form grants a lesser ability of the same type, you gain the lesser ability instead. Your base speed changes to match that of the form you assume. If the form grants a swim or burrow speed, you maintain the ability to breathe if you are swimming or burrowing. The DC for any of these abilities equals your DC for the polymorph spell used to change you into that form.
Reading Elemental Body, we have the following:
When you cast this spell, you can assume the form of a Small air elemental, Small earth elemental, Small fire elemental, or Small water elemental. The abilities you gain depend upon the type of elemental into which you change. Elemental abilities based on size, such as burn, vortex, and whirlwind, use the size of the elemental you transform into to determine their effect.
Air elemental: If the form you take is that of a Small air elemental, you gain a +2 size bonus to your Dexterity and a +2 natural armor bonus. You also gain fly 60 feet (perfect), darkvision 60 feet, and the ability to create a whirlwind.
As we can see, there is nothing replacing the more general text from the Polymorph subschool. Meaning that we use the DC of the spell used to assume the elemental form.
This can be seen on other spells, like Form of the Dragon's Frightful Presence ("DC equal to the DC for this spell").
Best Answer
Druids have been shapeshifters from the beginning
The D&D Druid ability to shapechange showed up originally in their first appearance in the game, as a monster in the Greyhawk supplement for original D&D. At that time, they were known as "priests of a neutral-type religion", had both cleric and magic-user spellcasting, and had "barbaric followers". "Powered by nature" does not appear to have been included at that time. They were later added as a class in Eldritch Wizardry, where the connection to nature was made more clear.
Gary Gygax himself has stated that the druid was based on Caesar's description of druids in Commentarii de Bello Gallico (at least, according to James Maliszewski of Grognardia). That description did not involve shapeshifting, but it does draw the connection between D&D druids and Gaulish priests of that era.
Near-contemporary to Caeser's work, Pomponius Mela, writing De situ orbis libri III in AD 43 or so describes a group of female priests of a Gaulish god. "They call them Gallizenae, and they believe them to be endowed with extraordinary gifts to rouse the sea and the wind by their incantations, to turn themselves into whatsoever animal form they may choose, to cure diseases which among others are incurable, to know what is to come and to foretell it." As those were likewise priests of Gaul, they'd also have qualified as druids.
We can't know for absolute certain that the D&D shapeshifting druid was inspired by any other source, but we can at least be quite sure that they weren't the first to come up with the idea.