You can be polite and unavailable at the same time.

Warmth and availability get bundled together, as though declining required being cold about it. They are separate things: you can be genuinely kind to someone and still not be free, and the kindness is what keeps the no from landing as rejection. A friendly refusal is harder to take offence at than a reluctant yes delivered badly. Your tone and your answer are allowed to disagree.

Practise the combination once this week. Say something pleasant, say you cannot, and add no reason. Then ask about them, so the warmth reads as real.