Drop the jaw noticeably. Keep the body of the tongue low and forward, and don't let the back of the tongue raise toward the soft palate. Pull the lip corners back slightly, almost a starting smile.

Americans pronounce add as AD (/æd/). You'll hear it in sentences like "Add the odd numbers" or "Add the odd code to the middle of the pad" — more examples below.
Record yourself saying "add" 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.
Drop the jaw noticeably. Keep the body of the tongue low and forward, and don't let the back of the tongue raise toward the soft palate. Pull the lip corners back slightly, almost a starting smile.

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

Click any sentence to see the full breakdown — every link, every reduction, every flap-T.
The textbook way isn't wrong — it's just not how anyone actually says it.
In "add", 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.