Learn Turkish: Ebeyi müjde verirken duydum. - I heard the midwife giving the good news.

In Turkish, perception verbs like duymak (to hear) or görmek (to see) take the person you are perceiving as their direct object. So, you aren't hearing 'the giving', you are physically hearing 'the midwife' (ebeyi) while she is giving the news.

Could I use an object participle here and say Ebenin müjde verdiğini duydum?

You could, but it drastically changes the meaning! Ebeyi müjde verirken duydum means you were physically present and heard her voice giving the news. Ebenin müjde verdiğini duydum uses a factual participle, meaning 'I heard (the fact) that the midwife gave the good news' — you likely heard this as gossip or second-hand information.

How would I apply this lesson's grammar to say I heard her right as she was about to give the good news?

You would take the imminent action reduplication verdi verecek (she is just about to give) and add -ken (while) to the end of it. The sentence becomes: Ebeyi müjde verdi verecekken duydum. This perfectly captures 'I heard the midwife right as she was on the verge of giving the good news.'

Why doesn't müjde take an accusative ending here like müjdeyi?

Here, müjde vermek acts as a general, unspecific action ('giving good news' in general, rather than 'giving the specific good news'). When a direct object is indefinite or part of a general verb phrase, it stays bare.