Each language has its own sentence structure and grammar. Although American Sign Language is native to the United States and can be quite different from sign languages of other countries, ASL and spoken English are not interchangeable. ASL has a different syntax from spoken English, and communication is not a word-for-word exchange. If you incorporate signs or signing into spoken storytelling, it would be confusing for a deaf audience member to see each word signed in the order in which it is spoken. It would come across a little bit like Yoda to someone who communicates through ASL.
For example The American Sign Language Phrase Book (Fant, Lou and Bernstein Fant, Barbara, McGraw Hill, 2008, p. 14,15) includes the signs for "Why didn't you eat last night?" In ASL, that could be signed, "past night you eat not why" or "past night why you eat not." In Signing the Scriptures, Year C (Blake, Joan, LTP, 2003, p. 13), Luke 2: 8-14 could be signed: "Near area, shepherds work watch all-night. Angel from Lord show-up, glory shine-down; they afraid! Angel tell-them, 'Afraid, nothing. I inform-you wonderful happen, good NEWS for ALL people. Today, recent there Bethlehem, born Savior, himself Messiah, Lord. Proof what? Manger, you go-to-it, notice baby, cloth wrap-up, lay-in.' Wrong, angels many! hordes sing praise God, quote, 'Glory give-to God heaven, peace touch earth for people he cherish.'"
Exact sign and word order is Signed English, or Signed Exact English (SEE). Although it is a combination of ASL signs, sometimes modified, and unique signs, it is not a language in and of itself. I don't have much experience with SEE and don't have specific recommendations to explore if you want to pursue that route. However there are no doubt plenty of YouTube examples.
If you are incorporating signs into spoken storytelling, I don't recommend trying to sign every word or necessarily signing words in the same order as they are spoken. However, selected signs can enhance meaning. You might also use a more word-for-word signing as a practice tool to help internalize the story. In other videos on this site, I work with a biblical passage and include some signs to accompany them. For me, these signs draw out an important point of the story or help me to see characters in more than two dimensions. As you are working with a story, choose which signs most enhance the message you are trying to convey, much as you would choose other movements, tone of voice, or perspectives.
Biblical storytelling, at least the NBSI* way, encourages high word accuracy and even higher content accuracy. Although the words used matter, paying attention to the message conveyed by the words (the gist of it) is where the heart and spirit of the story dwell. Honoring the words and the tradition through which they were formed keeps storytellers from making these sacred stories something of their own creation. It is a combination of words as they have been traditioned to us and the spirit that works through us as we tell a story that makes a message live in and through us.
*Network of Biblical Storytellers, International (www.nbsint.org)