Relax your lips and drop your jaw significantly. The tongue tip lightly touches behind the bottom front teeth and the back part of the tongue presses down a little to create more dark space in the back of the mouth.

Americans pronounce awesome as AH-suhm (/ˈɑsəm/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "We all thought it was an awesome opportunity".
Record yourself saying "awesome" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 4 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

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

The textbook way isn't wrong — it's just not how anyone actually says it.
In "awesome", the short unstressed vowel before "m" disappears — the schwa is absorbed and the "m" 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 AH — 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.