To make a long answer short: no, you should write like you write.
It doesn’t matter if words are spoken or written, language is driven by one purpose: to effectively communicate. In conversation, we are typically more casual in conveying information to others. Even in formal speech, there are certain liberties taken for creative expression.
The tools available to us for speech and for writing are different. Unlike speaking, writing does not come as easily. This might sound shocking to some, especially to writers. Look at it this way, as toddlers we learn to speak before we write. The written word must conform to general rules of organization, punctuation, and grammar. For emphasis, writing would omit repetition (or scarcely use it), but in speech, it is acceptable to some degree.
With words, you are left with other tools to transfer your message. You can’t use body language and facial expressions to help you. Adjusting the tone of language is dependent on your understanding of language.
Using a casual tone in your writing is acceptable — what it takes is using phrases and mechanics to create a conversational text. It does not warrant an excess of self-referential phrases (such as “as you will” or “so to speak”).
Casual text refers to simplistic and easy-to-understand writing. There is beauty in simplicity. The best way to put it, write like you are thinking, providing that you keep your thoughts organized. Remove the verbal ticks, the unnecessary repetition, and the stunted vocabulary.
Not everyone has a limited vocabulary, but when speaking, we tend to stick to words everyone knows. This doesn’t mean that your normal vocabulary needs to stay underdeveloped. To get your message across, be more selective of your words and their placements.
Some people can write as they speak — that is only if they speak technically correct. The majority does not speak with extreme caution: we speak in a relaxed manner. As a result, the way we think has also become relaxed. The only way to get in the habit of good writing is to practice.
One way to measure the weight of your words is to think about how you would react to them if they were said to you. Once you refine your words, you will refine your thinking.