Tongue pulls back slightly from the D position, blending into R. Sounds close to 'jr'.

Americans pronounce dressed as DREHST (/drɛst/). In "dressed", the "dr" cluster blends into a "jr" sound — a natural American English pronunciation. This is called the DR Sounds Like JR, a hallmark of natural-sounding American speech. It comes out as DREHST. You'll hear it in sentences like "The kids need to get dressed and eat before the school bus arrives".
Record yourself saying "dressed" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 5 sounds. Explore each sound's mouth shape and how it's made.
Tongue pulls back slightly from the D position, blending into R. Sounds close to 'jr'.

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 moderately. Touch the tongue tip behind the bottom front teeth and lift the mid-front part slightly toward the roof.

Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

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 "dressed", the "dr" cluster blends into a "jr" sound — a natural American English pronunciation. /d/ shifts toward /dʒ/ ("j"), so DR sounds like "jr".