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

Americans pronounce purred as PURD (/pɜrd/). You'll hear it in sentences like "The cat purred contentedly while sitting on her lap".
Record yourself saying "purred" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 3 sounds. 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.

Flare your lips and push them away from the face. Lift the middle of your tongue toward the roof of the mouth.

Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "purred", 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.
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.