More context might help but s’accrocher is used figuratively here and indeed means something like “to keep going”, “to hang” or “to cling” (thanks to aCOSwt and Laurent S. for the suggestions).

S’accrocher means “to refrain to give up, to keep going despite the difficulties”.

“S’accrocher” is literally similar to “hold on”, meaning “endure in difficult circumstances”, but its connotations are sometimes close to “keep going”. This sentence is difficult to translate and “keep going” just fits better the chosen English phrasing. Also note that in the French phrasing, reading the letters has more accurately become the only motive/purpose that keeps her going.