Tome Hayashi copes with her situation by writing poetry. That is to say, at least in the three-month period over which this story takes place. She is not just coping with her husband. She's also coping with a troubled past, one which she is running from but also trying to reconnect with. She writes in haiku, a historically Japanese form of poetry and she writes in Japanese. Living in America, Tome has a double consciousness which means she knows what it is to be Japanese (or Japanese-American) and American. This double consciousness is also shown in the two different people Rosie sees in her mother: Tome Hayashi and the writer, Une Hanazono.
The fact that she writes haiku in Japanese suggests a reconnection with her Japanese past, troubled as it was. The seventeen syllables of the haiku form are also an homage to her stillborn son who would have been seventeen by this time. Just as she adopted a new name (Une), this creative outlet gave Tome a way to become someone else in name and action. Even if it was an infrequent escape, it allowed her to dream. Winning the contest, she actually got public recognition that she'd achieved that dream.
You could also argue that Tome deals with being married to a "less-than-satisfactory" husband by trying to form a meaningful relationship with Rosie. However, this is problematic because of the generational gap and the cultural gap (Rosie was born in America). So, her relationship with Rosie is also frustrating.
Aside from writing haiku, the author doesn't provide any definitive answer as to how Tome copes. But like haiku, where the meaning is condensed into a short form, the author has left this information out, challenging the reader to come up with some possibilities. Who knows how Tome coped before and after her brief writing career? Perhaps she will now write in secret, maybe only sharing her poems with Rosie; that is, when she feels Rosie is mature enough to understand why she needs to write.