how the hell am I this far into vocab learning and the verb apologise has never turned up???

Heh, I guess 对不起 is the too-obvious tick for the "sorry" box…?

In a similar vein of various directions you get:

• 致歉 when you're 道歉ing but more formally
• 抱歉 as in (recorded phone voice) “很抱歉，您拨打的用户暂时无法接通…”
• Also, similar constructions: 道别, 道谢…