Expression : ' I went off to sleep.'

[Tom]

Hi

Could you please tell me another natural way I can say the following (red)?

[Yankee]

Hi Tom

Do you mean the actions (read and sleep) to be two very clearly separated actions, or did sleep interrupt your reading?

For the second scenario, you could say these:

I nodded off/fell asleep/dozed off (while I was) reading my book.
.