(혜진씨한테) 문자를 보냈어요+약속이 바뀌었어요= (혜진씨한테) 약속이 바뀌었다고 문자를 보냈어요 or (혜진씨한테) 약속이 바뀌었다는 문자를 보냈어요 편지를 보냈어요+시험에 합격했어요= 시험에 합격했다고 편지를 보냈어요= 시험에 합격했다는 편지를 보냈어요

Hi :) I think I'm going to help you. Actually, as a native speaker, both are okay. But If you want to use the second one you should change a little bit "혜진씨한테 방금 약속시간 바뀌었다는 문자'를' 보냈어요". Cheer up!

I'm not a native speaker, but the forms are just multiple ways of using conveying indirect statements. It feels much more like a book or news report stating facts when I see the 다는/라는 ... forms and more colloquial/an opinion when I see ~다고/라고. Also, the whole form of ~다는 ... is really ~다고 하는 ... I think that if the context was clear to the listener in this context, the whole ~다고 방금 문자 보냈어요 part of the sentence could even be shorted to ~대(요). It would probably be the result if someone had been wondering where 혜진 is. Also, I think that in your second sentence, you'd need to move the word 방금 because in that location, it seems to be describing the time change rather than the message was just sent. Hope this is kinda of helpful :p someone else could probably give you a better answer

I guess it sounds better like that in English, while it sounds more natural the way it was written, in Korean.