Touch the front of your tongue to the roof of your mouth, then release into a 'sh' position. Flare your lips.

Americans pronounce chores as CHORZ (/tʃɔrz/). You'll hear it in sentences like "The store bore a score of poor chores".
Record yourself saying "chores" 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.
Touch the front of your tongue to the roof of your mouth, then release into a 'sh' position. Flare your lips.

Start with the 'aw' jaw drop and rounded lips. Pull the tongue back and up while keeping the lips rounded for the R.
Same position as S, but add vocal cord vibration. Feel the buzz.

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