1A humble (kenjōgo) form of "to know" — used to modestly state that one is aware of or acquainted with something, typically in formal speech or writing. A somewhat more classical/formal counterpart of the more common 存じる.
皆様のご健勝を心よりお祈り申し上げると存ずる次第でございます。
I humbly and sincerely wish everyone continued good health.
その件については、私も多少存ずるところがございます。
Regarding that matter, I too am aware of some details.
お名前はかねてより存ずる方でございます。
You are someone whose name I have long been acquainted with.
2A humble (kenjōgo) form of "to think" or "to consider" — used to modestly express one's own opinion or judgment, most often in the fixed pattern 〜と存ずる in speeches or formal documents.
この計画は実行可能であると存ずる。
I believe this plan is feasible.
今回の処置は妥当であったと存ずる次第です。
I am of the view that this measure was appropriate.