Drop the jaw noticeably. Keep the body of the tongue low and forward, and don't let the back of the tongue raise toward the soft palate. Pull the lip corners back slightly, almost a starting smile.

Americans pronounce avenue as A-vuhn-yoo (/ˈævəˌnju/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "The office is located on Park Avenue".
Record yourself saying "avenue" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 5 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Lift your bottom lip so its inner edge (where the wet part meets the dry part) touches the very bottom of your top front teeth. Add vocal cord vibration as you blow air through.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
The schwa before N disappears — N becomes the vowel of the syllable. Go straight from the previous consonant to N.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "avenue", the short unstressed vowel before "n" disappears — the schwa is absorbed and the "n" becomes the syllable nucleus on its own. Schwa is absorbed — consonant becomes the syllable nucleus.
Stress falls on the first syllable, not the others. Stretch A — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.