You can say "I shower in the morning" or "I showered this morning." To say "I shower within the morning" is the wrong use of the word within. "Within" is used to say that something happened in a set amount of time or less, and that it will pass before something happens. "In" is used to say that a specific set time will pass before another action or event occurs. To put it more plainly, there are a few examples below:
I will be there in ten minutes --> I will arrive in exactly ten minutes (the set amount of time).
I will be there within ten minutes --> I will arrive in less than ten minutes.
To use within in your sentence, you are saying that you showered in less time than the morning, which doesn't really make sense. To say that you showered in the morning sets a specific time in the day, but doesn't convey anything less or more.
I hope this makes sense. It is quite a difficult topic to explain, and also challenging to grasp at times, even as a native speaker. Let me know if you have any other questions!