What should I do if I gave money to a woman to distribute to the needy, and I have doubts that she kept it for herself? Should I ask her for the beneficiaries' numbers to verify, or should I leave the matter to her conscience and her accountability to Allah?

If you wish to appoint someone as an agent for charity (sadaqah (definition)), choose someone you trust. If you suspect her conduct, you can dismiss her and reclaim the money before it is spent, or ask her where she spent the money and verify it, especially if there is strong circumstantial evidence of her betrayal. However, if there is no such evidence, the presumption is that she is trustworthy, and there is no need for verification. If her betrayal is proven, she is accountable for the money and must return it.