The Creative Commons Attribution-ShareAlike 2.5 license, like all the creative commons licenses, applies only when a person wants to do something that required permission under copyright law. The license grants that permission, but only if one complies with its terms.
If a person wants to do something for which no permission is required under copyright law, then the licence is irrelevant and the person does not need to comply with its terms.
Version 2.5 of the CC-BY-SA licence expressed this concept in section 2. Fair Use Rights, which reads:
Nothing in this license is intended to reduce, limit, or restrict any rights arising from fair use, first sale or other limitations on the exclusive rights of the copyright owner under copyright law or other applicable laws.
The CC Wiki page on License Versions reads:
Exceptions and limitations unaffected
All CC licenses only govern uses that would otherwise be restricted by copyright and other closely related rights as provided in the licenses. If a use is not regulated by virtue of an applicable exception or limitation, the license does not apply and there is no need to follow the license conditions. The licenses do not create obligations where they would not otherwise exist.
Version 4.0 of the CC-BY-SA license includes a Considerations for licencors, which reads:
Our public licenses are intended for use by those authorized to give the public permission to use material in ways otherwise restricted by copyright and certain other rights.
Section 1.f of the license defines "Exceptions and Limitations":
Exceptions and Limitations means fair use, fair dealing, and/or any other exception or limitation to Copyright and Similar Rights that applies to Your use of the Licensed Material.
Section 2.a.2 of the license reads:
Exceptions and Limitations. For the avoidance of doubt, where Exceptions and Limitations apply to Your use, this Public License does not apply, and You do not need to comply with its terms and conditions.
The question reads:
Let's say I make a documentation-like website using information that came from articles licensed under the [CC-BY-SA 2.5 license] ... I will be copying parts of the original article, but I will re-word and re-organize everything.
If this "copying" goes beyond the use of ideas from the source work, and beyond what is permitted under fair use (see 17 USC 107), then permission is required. In that case permission has been granted only under the terms and requirements of the license (or "a later version of this License with the same License Elements as this License" which would mean CC-BY-SA 3.0 or CC-BY-SA 4.0 at this time). In this case you must place your entire derived work under the same license, or a later version of the same license.
If however, you only copy facts and information but not any details of expression, or only copy expression to the limited extent permitted by fair use, then the result is not a derivative work, and the license does not apply. In this case you do not need to place the resulting work under a CC license of any sort, although you may choose to do so. Nor are you obliged by the license to provide attribution, although fair use may require some degree of attribution, and many people would say that there is an ethical duty to provide it.
In short you may not publish a derivative work from the CC-SA licensed text (or other work) unless you "share alike" by using the same license (or a later version of it). If a work is not derivative but has only been influenced by repeating factual information, in a significantly different expression, then there is no legal requirement to "share alike".