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

Americans pronounce proudly as PROWD-lee (/ˈpraʊdli/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "Proudly announce the outcome of the round".
Record yourself saying "proudly" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 6 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.
Start with a dropped jaw and flat tongue. Glide into a relaxed, slightly rounded lip position as the back of the tongue stretches up.
Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

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 "proudly", the "d" 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 PROWD — keep everything else short and quick.