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

Americans pronounce promptly as PRAHMPT-lee (/ˈprɑmptli/). In "promptly", the "t" is squeezed between other consonants and drops out — the surrounding consonants flow together without it — most natural in flowing, casual speech; in careful or formal speech, the T may be lightly present. This is called the Silent T in Clusters, a small move that separates 'classroom' from 'native'. It comes out as PRAHMPT·lee. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "Please pay the parking permit price promptly".
Record yourself saying "promptly" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 8 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 to stop the air, then release. No vocal cord vibration.

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.
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.

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

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

The T is skipped entirely. Your tongue doesn't make contact at the T position.

Place the tip of your tongue against the alveolar ridge just behind your top front teeth, the same contact point as /t/, /d/, and /n/. The difference is what happens to the air: for /l/, you let it flow continuously around the <em>sides</em> of the tongue (that's why /l/ is called a lateral). Turn your voice on the whole time. Lips stay relaxed, no rounding or flaring. For the Dark L variant at the end of a syllable, also pull the back of the tongue up and back toward the soft palate.

Pull the corners of your lips back slightly. Arch the middle-front of your tongue high toward the roof of the mouth.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "promptly", the "t" is squeezed between other consonants and drops out — the surrounding consonants flow together without it — most natural in flowing, casual speech; in careful or formal speech, the T may be lightly present. /t/ is dropped entirely — the surrounding consonants flow together without the T.
In "promptly", 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 PRAHMPT — keep everything else short and quick.