"Check that" conveys the confirmation of a presumed fact or condition.  "Check if" contains no such presumption.

If lack of presumption is explicit in the sentence, "if" must be used.  Otherwise, either conjunction may be used without loss of grammatical correctness.

>I checked that he was awake.  *(presumes a given condition)*

>I checked if he was awake.  *(may or may not include presumption)*

>I checked if he was awake yet.  *(lack of presumption is indicated by "yet")*

>I checked that he was awake yet.  ***(grammatically incorrect)***

When the object of "check" is a standalone sentence, using no conjunction will seem clumsy to native English listeners and may slightly obscure the sentence's meaning (e.g., the OP's first example sentence in a spoken context).