Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Americans pronounce core as KOR (/kɔr/). You'll hear it in sentences like "The core corps wore the torn uniform" or "Four more doors explore the core floor" — more examples below.
Record yourself saying "core" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 2 sounds. Explore each sound's mouth shape and how it's made.
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.