C: Sentonte dormemon, li injektis la drogon.

But for some reason this sounds odd, and I can't explain why

This is odd because generally you don't know what your future state of alertness will be. And it is not the equivalent of C: Li injektis la drogon kaj [baldaŭ] poste li sentis dormemon which is - he injected and then felt sleepy'

You could say however:

Devonte senti sin freŝa dum la sekva tago, li injektis la ...

Because he knew it was expected of him to be well-rested on the following day, he took a sleeping pill