Of course every word can be used in spoken Chinese as is the case in English ^_^ but u r right it's a pretty formal word and it would sound a bit off if u say it to someone's face . Plus I think this word usually occurs in literature work . Cuz normally when you wanna show your apology to someone , you just say ,对不起 or 非常抱歉. Or in written language , if u wanna come off as more formal , you. Could probably say 请接受我诚挚的歉意 or something like that . 赔罪 sounds more acceptable in ancient times or in literature work . A similar term is 赔礼道歉 I think it's acceptable in modern spoken Chinese , but it indeed sounds very formal in oral language , for example , "我必须要亲自上门给他赔礼道歉。” hope it helps !

Colloquial 赔礼 < 道歉 < 赔罪（<谢罪） Literary 赔礼、道歉 are common words in both spoken and written language. 道歉 is the most frequent. 赔罪 is more used in literature, but is ok in oral expression. 谢罪 is an old expression, and rarely used today( but it's a common word in Japanese).

对不起 and 抱歉 are more colloquial but can sound cavalier. 赔罪 is more formal and shows greater sincerity. It is akin to (without context) , "so sorry" vs "I am so sorry".

I don't think I've ever said this word.

In general，道歉 = 赔罪。赔罪 is very native~ When you want to apologize to somebody you can say:我给你赔罪啦。