Cantonese is definitely the most widely written Chinese language after Mandarin, but there are also others. Obviously they are not as healthy, due to traditional social pressure to use formal Chinese in a written context, and political pressure from government(s) to use 普通話/國語 in all contexts. Taiwanese, for example, is most often 'written' in the manner you describe -- the spoken sentence is composed, and then written in what the speaker believes is an appropriate Mandarin equivalent (if in a formal context, such as subtitles, it is usually translated into a proper mandarin sentence; for cases like song lyrics, it is given approximation in Chinese characters).

However, there are absolutely other forms of written Taiwanese, most popular being POJ romanization, but also forms based on Bopomofo, as well as character based methods, including those promoted by the MOE amongst others (see http://twblg.dict.edu.tw/holodict_new/default.jsp ; various museums, especially http://www.nmtl.gov.tw/ ; any number of Taiwanese dictionaries, or books with titles along the lines of 正字, and so on). There are standards, which have unfortunately suffered due to infighting and lack of government support in schools, but standards nonetheless; while the number of Taiwanese people that use them is very small and mostly academic, they do still exist, and there are people who write directly in Taiwanese, whether via romanization, Chinese characters (including Chinese characters invented specifically for use in Taiwanese), or Taiwanese bopomofo. [Honorable mention: Taiwanese kana; also, though written works tend to be in Mandarin, it is worth noting that many shops throughout Taiwan use Taiwanese on their signage]

'Written dialects' is what everyone does in Chinese now, only with Mandarin as the only one in a position of power. Originally, Chinese wasn't invited to 'translate across dialects'; there was a prestige language, which turned into what we know as Classical Chinese, which was written and propagated. Later, as both the characters and the language (which was now separate from the spoken language, as Latin separated from the romance languages) were extremely difficult to use, there was a push towards 白話字; writing representing spoken word. This eventually succeeded, though much more so with prestigious, more widespread languages (Mandarin and Cantonese). Classical Chinese didn't 'translate across dialects' any more than Latin 'translated across' French and Italian; it was merely the written lingua franca of the time. An educated man would learn to read Classical Chinese, through his own language, sure, but not as a written form of it. The idea that you can write any Chinese language down and read is in another is a popular misconception, but a misconception as you well know from your attempts to write Taiwanese, or read Cantonese.