Lift your bottom lip to touch the very bottom of your top front teeth. Blow air through this contact point without voicing.

Americans pronounce fans as FANZ (/fænz/). In "fans", the "a" vowel before M or N raises and fronts toward [eə] — the tongue pulls up and forward, breaking the vowel into a tense glide as it anticipates the nasal. This is called the Cat-Vowel Before M/N, a hallmark of natural-sounding American speech. It comes out as FANZ. You'll hear it in sentences like "The stadium was packed with cheering fans wearing team colors".
Record yourself saying "fans" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 4 sounds. Explore each sound's mouth shape and how it's made.
Lift your bottom lip to touch the very bottom of your top front teeth. Blow air through this contact point without voicing.

The tongue relaxes down in the back and the corners of the lips relax before the consonant. This adds a schwa-like 'uh' relaxation after the /æ/. Think of it as 'relaxing out of the vowel' — it is no longer a pure /æ/ sound.

Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

Same position as S, but add vocal cord vibration. Feel the buzz.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "fans", the "a" vowel before M or N raises and fronts toward [eə] — the tongue pulls up and forward, breaking the vowel into a tense glide as it anticipates the nasal. The "/æ/" vowel raises and fronts before M or N — tongue pulls up and forward, producing a tense [eə] glide (between /e/ and /ə/). Not a pure /æ/.