Push a stream of air from your throat through your open mouth. No tongue or lip contact.

Americans pronounce hammer as HA-mer (/ˈhæmər/). In "hammer", 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, and it's one of the defining features of casual American English. It comes out as HA·mer. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "He hid the heavy hammer behind the hedge".
Record yourself saying "hammer" 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.
Push a stream of air from your throat through your open mouth. No tongue or lip contact.

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.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "hammer", 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/.
Stress falls on the first syllable, not the others. Stretch HA — keep everything else short and quick.
Americans use a relaxed retroflex R — the tongue curls back rather than rolling. The R is one continuous sound with the vowel before it, not two separate sounds.