I think I see what your problem is. According to my handy grammar handbook:
The exact words of a speaker should be set off from the rest of the sentence by using a comma, a question mark, or an exclamation point.
Note: If the quotation is a grammatical part of the writer's sentence, the quotation should not be set off by commas, nor should it begin with a capital letter.
~Abeka God's Gift of Language B
So what does this mean? If your quotation is actually the character speaking, it is a direct quotation. You use commas to separate it, and it starts with a capital letter (do note that if the quotation is interrupted and then continued, the continuation does not start with a capital letter).
However, if your quotation is is not a direct quotation - that is, you are describing what was said rather than the character actually saying it - then you do not use commas.
Below I'll correct your examples and explain each one.
I cover the receiver and whisper, “Sorry,” to my friend.
This is a direct quotation, because someone is saying it. Therefore, you use a comma, and it starts with a capital letter. There should also be a comma at the end of the quotation.
He leans forward and whispers, “I’m ready, baby,” into my ear.
This is correct. It is what he is saying, making it a direct quotation. However, you should have a comma after baby, as I added. Because it is the beginning of a direct quotation, it would also begin with a capital (which it does anyway because it starts with 'I')
I resist the urge to shout "screw you, asshole" right to his face.
The quotation is not actually said, meaning it is not a direct quotation. Therefore, commas are not used, and your example is correct. A comma is not needed after the quotation either, because it is not direct.
I turn to him and mouth "bye."
This is a tricky one. I am actually not 100% sure on this, but I believe because nothing was actually said, it is not a direct quotation, and therefore would be correct the way you have it. Do note that all punctuation goes inside the quotation marks, even if it isn't part of the quotation. Observe the period I've added above.
I try to shout "stop" again but the words won't come out.
I feel extremely tempted to say “just drive” and hope for the best.
Once again, nothing was actually said, meaning that both of these sentences are perfectly correct. Note that I made 'just' with a lowercase 'J,' because it is not the beginning of a direct quotation.
So a good rule of thumb to follow would be to determine if something is a direct quotation or not. If the quotation is what someone actually said, then it is, and you use commas and a capital. If it is what someone wanted to say, but didn't, or anything similar, it is not a direct quotation, and quotation marks alone will suffice.