Speaking from a script doesn't have to sound scripted, I neglected to mention a big red flag that you're sounding stilted and stiff: Missing contractions.
As writing is a more formal way to express ourselves, we tend to write without contractions. Therefore, in writing I might say, "I am" instead of "I'm" and "You are" instead of "You're."
However, if you speak like you write -- in this case, reading the full expression instead of the shortened one -- you come across as overly formal. That's not to say that you can't mix and match contractions with the written-out version of expressions. But speaking entirely without contractions sounds unnatural and will certainly peg you as reading straight from a script, in addition to becoming a distraction for the audience.
A perfect example of the awkward formality and distraction of spoken language without contractions was the recent remake of the movie "True Grit." The language in the movie is not contractionless, but it is pretty heavy on uncontracted words.
Watching this movie, you might have come to the conclusion that contractions were not a normal part of speech (at least by mercenaries and lawmen) in the late 1800s (which is how the language in the original True Grit novel by Charles Portis was portrayed, and then repeated by the Coen brothers in their version of the movie). This article, by Mark Liberman, disputes this and concludes that this dialogue was more an artistic choice than a faithful representation of 19th century speech.
Contractions comes naturally, it's customary in the English language to use them, and it's the way others expect to hear us speak.
If you're writing a speech, go through it sentence by sentence and see where you've left out contractions. You might find that leaving some words uncontracted allows for a stronger emphasis. If it makes sense, don't contract. Otherwise, make sure to convert most of your formal phrasing into contractions. And keep reading the speech aloud as you go to make sure you sound natural.
Remember: Written and spoken language are not the same thing. Make sure that, if you're going to be reading from a speech, that you've made sure to keep your language as natural-sounding as possible. Contractions have been around for hundreds of years, and they're here to stay, so use them!