Listen to the difference:
Oh I díd. «Don't worry, I already did.»
Óh, I díd. «Thank you for reminding me! I already did it.»
Alex: Dad, don’t forget to tell mom I need $50 for next week’s school trip.
Dad: Oh I did. I told her this morning. She says she’ll give it to you tonight.
In this cartoon, the guy assumes the dog is waiting for its dead owner to come back, but the dog uses oh to assure him that he's doing no such thing — he's just there to remember the times he had with his owner.
For instance, in this snippet below, Fred's oh has a very smug quality. You can picture him smiling smugly with his arms crossed. He's not only telling his wife he already knows about the party, but in using this oh, he's expressing a sarcastic assurance, preparing his wife for the bombshell that her friend, Barbara, isn't as discreet as his wife thinks.
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This type of reassurance can also be expressed by the phrase Don't worry,... which, in this usage, doesn't literally mean "don't worry," but rather «you can be sure of this». So, in the snippet below, Fred could also have said "Don't worry — I know."
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Wife: Fred, I’m throwing a surprise party for your sister next weekend. But don’t tell anyone — nobody knows about it except you, me, and my friend, Barbara. But she won’t tell anyone.
Husband: Oh I know.
Wife: You know about the surprise?
Husband: Oh yeah.
Wife: How??
Husband: Your friend, Barbara, told me.
Here, the nuance of «don't worry» becomes a more emphatic «Don't you worry!» or even «You can be (damn) sure of that!». Note that all right is pronounced with a low intonation.
Carla: Have you noticed how Sandy looks at Cooper during meetings. If I didn’t know better, I’d swear they were together.
Nancy: Oh they’re together all right — I saw them making out in the parking lot this morning before work.
Nicky: What’s this, a C in science? A D in Spanish? That was your best subject last year.
Natasha: That’s because he plays basketball with Alex every day after school and then bangs out his homework as fast as he can so he can play video games.
Danny: Sorry! I’ll bring my grades up!
Nicky: Oh you’re going to bring your grades up all right… you’re going to come straight home after school every day and do your homework, AND I’m getting you a Spanish tutor.
3 Unstressed Oh is commonly used in front of something you want to play down or make sound less important or even unimportant in order to sound humble and unassuming.
The humor in this meme lies in the fact that someone has obviously just asked this guy how he stays in such good shape. In response, he flexes his right arm to show off his biceps, while making a humble, dismissive, waving motion with his left hand to play down his exercise routine (obviously, nobody gets that kind of physique by just "jogging every once in a while".) Here, the initial Oh just complements the dismissive gesture and makes his response sound even more humble.
4 In front of a command, Oh is used to express impatience and frustration.
In this usage, it's always used with just (also unstressed).
In this meme, the speaker is so tired of reading stupid Facebook statuses that he feels like telling the author to just shut up! He feels like he can't read one more.
Brad: Can I have the wifi password?
Jimmy: It's my birth year...
Brad: 2009
Jimmy: Plus my house number...
Brad: Oh just give me the password already!