mostly yes; however there are some differences
Only söylemek is used when;
telling someone to do something
Bana gelmemi söylüyor. - He tells me to come.
Ayşe bilmediğini söylüyor. -> Ayşe says she doesn't know.
Only demek is used when;
"I don't know" Ayşe said. -> Ayşe 'bilmiyorum' dedi. or Ayşe dedi ki "bilmiyorum".
This is very helpful, but I'm confused about the difference between indirect speech ("Ayşe says she doesn't know") and "ki" sentences ("He said that this is wrong"). Is "ki" used for much other than indirect speech? Can you change an indirect speech sentence with "ki" to one without by simply removing "ki" and changing replacing "demek" with "söylemek"?
Well, I just edited and made it more clear. The 'ki' version 'dedi ki' is actually a type of direct speech. So you have two options doing direct speech: putting 'dedi ki' at the beginning of the sentence or putting 'dedi' at the end of the sentence. Both are equally frequent.
The thing is that direct speech is much more frequently used than indirect speech in daily speaking. This is naturally only a case for demek and söylemek.
In other cases we don't use ki sentences much.
Gittiğini fark ettim. - I realized that he was gone.
Fark ettim ki gitmiş. - I realized that he was gone. (less used)