Are non-Mandarin Chinese speakers offended by referring to Mandarin Chinese as simply "Chinese"? Is that a politically-fraught term?
As a native Cantonese speaker, I do not object people referring to Mandarin as 'Chinese' because it is the official language of Mainland China, and Most Chinese do live in Mainland China
On the other hand, Cantonese and other Chinese dialects are Chinese too. Canadian Government treat Mandarin and Cantonese as equal. When Canadian ask you "do you speak Chinese" and get an answer "yes", they do not automatically presume you are a Mandarin speaker, they would also ask " Mandarin or Cantonese?" because the population of Mandarin and Cantonese speakers are roughly the same here.
How do I differentiate between the dialects when speaking/writing in Chinese?
Dialect is a spoken language, there are hundreds of different dialects in China, most of them are not mutually intelligible. It is hard to differentiate between minor sub-dialects even for Chinese, I can only tell the difference between Major dialects like Mandarin and Cantonese and some Cantonese sub-dialects
All Chinese write in one standard language system. (Standard Written Chinese - SWC)
People who speak different dialects in China are expected to communicate with each others using the official dialect Mandarin in speech and the standard written Chinese (SWC) in writing.
Base on this practice, foreigners should be able to presume someone who speak "Chinese" can speak Mandarin, therefore, referring Mandarin as Chinese should not offend non-native Mandarin speaking Chinese.