Here is the paragraph from a textbook:
One day the Princess decided that she didn't like staying at home all day. so she told her father that she wanted to get a job.
My question is: can I replace didn't like with doesn't like, and wanted with wants so the paragraph becomes:
One day the Princess decided that she doesn't like staying at home all day. so she told her father that she wants to get a job.
No, it's not possible to make that change, although there are similar situations where it might be possible.
To make things clearer, here is an example where there is certainly only one correct choice:
Yesterday, when she saw her father, she told him she was hungry.
It is definitely impossible to say is here, because the state of her being hungry is in the past. This is the opposite of what is done in Slavic languages, where the equivalent of is would be required, because the tense in reported speech is determined from the point of view of the person at the time they're speaking. In English, as in most Germanic and Romance languages, it is the point of view of the narrator that matters.
There are cases in which things are not so clear-cut, however. Here is one case where you have a choice:
When I met her yesterday, she said that oxygen was/is heavier than carbon.
Here, even from the narrator's present perspective, there are two ways of looking at the situation. The first is to consider the statement "Oxygen is heavier than carbon" to be a timeless truth, the time at which the statement was made therefore being irrelevant. This corresponds to the use of is. The other possibility is to follow the general rule illustrated above even though in this case the statement refers to a timeless truth.
Here is another situation where you have a choice:
When we spoke yesterday, she said she would be/will be going to Ottawa next week.
(In either case, the words "next week" refer to what would be called "next week" relative to the present, not relative to yesterday.) Here it is again irrelevant whether we view the statement as having been true yesterday when it was made, or as being true now. There is no suggestion that the speaker might have changed her mind, or would say anything different now. Hence both tenses are possible. If the narrator uses "will be," then he is presenting the information as being equivalent in a practical sense whether viewed from the present or from the time it was originally conveyed. Perhaps the most cautious thing to do, if there was any doubt, would be to use would (which is the tense we use for the "future in the past").
In the following situation, only would is possible.
When I saw her this morning, she said she would have lunch with me tomorrow. But soon afterwards she phoned to cancel.
To return to your example, here is another sentence that is possible.
Yesterday Jane told me she doesn't like staying home all day. She is looking for a job.
However, in your example, the princess is a character in a story that is being narrated in the past tense. The princess doesn't exist in the present, so it would make no sense to say that she "doesn't" like staying home all day.
You can, for reasons of style, if you also change the other verbs there too to the present tense:
One day the Princess decides that she doesn't like staying at home all day, so she tells her father that she wants to get a job....
EDIT:
There is more laxity in how you phrase things, when you're telling a story. But you can't have jarring changes. For instance you could have the intro in the past but then starting with a new paragraph come to the present and stay there. Present tense brings the readers into closer contact with the story, so it's an interesting choice, and there is a tradition behind it too, so readers won't be shocked by that change. However, then you got to stick to it. Also, be careful with your punctuation.
