Press your lips together. Air flows through your nose. Vocal cords vibrate.

Americans pronounce manuscript as MAN-yuh-skrihpt (/ˈmænjəˌskrɪpt/). In "manuscript", 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, the kind of sound shift that makes everyday speech feel effortless. It comes out as MAN·yuh·SKRIHPT. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "I submitted my manuscript for publication last month".
Record yourself saying "manuscript" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 11 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Press your lips together. Air flows through your nose. Vocal cords vibrate.

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.

Lift the middle of your tongue toward the roof of your mouth, but stop just short of touching. /j/ is an approximant, not a stop. The tongue tip stays down, lightly resting near the back of your bottom front teeth. Voice runs through the whole gesture, and the tongue glides smoothly down into the next vowel. The lips stay neutral or pre-shape for the upcoming vowel (rounding early for OO in <em>youth</em>, for example).

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Curl or bunch your tongue without letting the tip touch the roof of your mouth. Brace the sides of your tongue against your upper back teeth, and round your lips slightly.
Drop your jaw slightly with relaxed lips. Touch the tongue tip behind the bottom front teeth and arch the top-front toward the roof.

Press your lips together to stop the air, then release. No vocal cord vibration.

Touch the tip or front edge of your tongue to the roof of your mouth just behind your teeth. Keep your jaw relaxed. Stop the air, then release with a puff.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "manuscript", 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/.
In "manuscript", the "t" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.
Stress falls on the first syllable, not the others. Stretch MAN — 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.