Him/to give up/of/bad/smoking/everybody/promise/this/heard/habit correctly position

Everybody heard him promise to give up this bad habit of smoking.

Since Everybody is capitalized, that's a clear indication it is the first word of the sentence. The main verb of the sentence that agrees with everybody is heard. Smoking doesn't work because it would require a helping verb, and there are none. We know from various studies and required warnings by the Surgeon General that smoking is a bad habit.

The word in the sentence that is an adverb is "today".

Explanation:

"Today" is the adverb in the sentence. It is an adverb of time.  Adverbs of time tell us when an action happened, but also for how long, and how often. When you put an adverb that tells us when an action happened at the end of the sentence, that is neutral position. But adverbs can be placed in different positions for the purposes of emphasis. For instance, in this case, it is in initial position in the sentence. Consequently, this time element is emphasized.