Is it permissible to pay the expiation for the days I missed fasting in Ramadan—unintentionally, not deliberately—to a governmental charitable institution that undertakes the care of orphans, the needy, the poor, and people with special needs?

Expiation (kaffarah) is only obligatory for breaking the fast by intercourse, according to the preponderant opinion. As for breaking the fast by means other than intercourse, it only necessitates making up the missed day (qada'). If you investigated the times for beginning and ending the fast and no error became apparent to you, then there is nothing upon you. However, if you ate thinking that the night remained, but then dawn became clear, or thinking that the sun had set, but then its non-setting became clear, then making up the day is obligatory according to the majority of scholars, and this is the more cautious approach. The same applies if you ate while doubting the sunset. If making up the fast becomes obligatory upon you, then ascertain the number of days and make them up. Nothing else is required of you, nor are you required to do anything other than to act upon your investigation and your strong presumption. The fidyah (ransom/expiatory feeding) becomes obligatory upon you if you intentionally delay an obligatory make-up fast while knowing of its impermissibility. This fidyah is feeding one poor person for each day you delayed, and you should give food, not its monetary value.