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feeling (much) obliged; being (very) grateful; being thankful
feeling sorry (for troubling someone); feeling ashamed (e.g. of an error); feeling embarrassed (e.g. by someone's compliments, hospitality, etc.)
お手数をおかけして大変恐縮ですね。
I apologize for the trouble.
恐縮ですが、少々お待ちください。
I'm sorry, but please wait a moment.
このようなお礼を頂いて恐縮です。
I'm humbled to receive such a gift.