Is it obligatory to exercise caution and search for those deserving of the funds that my mother used to distribute from charities and zakat, or should I take the initiative to distribute them for fear of the saying, "The soul of a believer is suspended by his debt until it is paid off"?

The agency is invalidated by the death of the agent. Since your mother passed away, her agency in paying charities has become void. You must return the funds to their owners, who can then dispose of them as they wish, and they can appoint you as their agent anew. Your mother's liability is cleared of these funds because they are not debts owed by her.