Word that means "to fill the air with a bad smell"?

"The air was still rank with the smell of dead animals." Rank in this case meaning an offensive smell or taste, especially one associated with decay.

Although this word does not fit exactly into your sentence, I'd like to suggest using the word reek to hint at the same meaning.

The air reeked of dead animals

According to Oxford Dictionary, reek means

Smell strongly and unpleasantly; stink

The air was still foul with the smell of dead animals.

foul

1 offensive to the senses, esp. through having a disgusting smell or taste or being unpleasantly soiled : a foul odor | his foul breath.
...
3 containing or charged with noxious matter; polluted : foul, swampy water.

New Oxford American Dictionary

Here are a few usage examples from Google Books:

Even though we were at least ten miles out in the muddy Loire estuary, the air was still foul with the stench of burning oil from the docks in St Nazaire.

Here, the heat was contained within a cavernous building, where the air was foul with the smell of coal smoke and machine oil and unwashed bodies.

The air was foul with their stench, although there seemed to be plenty of ventilation of a kind and the tunnel was dry.

Our artillery must have killed Japanese there earlier, because the air was foul with the odor of rotting flesh.

"The air was still "heavy" with the smell of dead animals.".

This definition of "heavy" is not exactly intuitive to non-native speakers but is generally idiomatic.

Definition of "heavy" from MWO:

g: oppressive (a heavy odor), (heavy weather), (rule with a heavy hand)

How about "fetid?" (I believe I have seen it spelt "fœtid," as well.)

Definition of fetid : having a heavy offensive smell
fetidly adverb.
fetidness noun.

merriam-webster.com