St., like Mt. or Ft. or less commonly Ctr., Spgs., Hts., and so on, are just abbreviations. While it is less common to see Sault Sainte Marie as opposed to Sault Ste. Marie, it is not "incorrect" to write it out as such, and for the most part, you can expect official laws and documents to write out the name in full.
How you should represent the name of a place is a matter of style. Adhere to the discipline of your editor, publication, or organization, or in the absence of a house style, adopt a style manual appropriate to your audience and tastes and be consistent in its application.
According to the Chicago Manual of Style, for example,
Prefixes of most geographic names should not be abbreviated in text:
Fort Wayne Mount Airy South Orange
Saint Cloud San Diego Port Arthur
Many editors make an exception for names beginning with Saint (St. Louis, St. Lawrence), and where space must be saved any such prefixes (except San, Santa, etc.) may be abbreviated (Ft., Pt., Mt., S. [South], etc.).
For an alternative view, the AP Stylebook (2013) states
saint Abbreviate as St. in the names of saints, cities and other places: St. Jude; St. Paul, Minn.; St. John's, Newfoundland; St. Lawrence Seaway
But see the entries for Saint John and Sault Ste. Marie
Saint John The spelling for the city in New Brunswick, to distinguish it from St. John's, Newfoundland
This contrasts with the AP guidance for fort and mount, which are never to be abbreviated.
These do suggest that for a large proportion of American writing, at least, St. is a much more common and acceptable abbreviation than the others in running text.
A single Wikipedia article will not be a reliable guide to usage. The local newspaper is the St. Paul Pioneer Press—
and the abbreviated form is also used by the state tourism board, the airport, the independent baseball team, and so on. For that matter, the Wikipedia discussion on the name points out that both St. Paul and Saint Paul have widespread usage, and "It's almost always useless to debate orthography in this manner."