I don't have a problem with "the cancer". "Her cancer" is also correct but I think it is redundant.
I don't really like the latter half of the sentence as you have written it. First of all, "wore" should replace "was wearing" to stay consistent with the previous verb "fell".
I have two suggestions for the sentence as a whole.
1. Split the sentence into two sentences.
During the cancer treatment, all Gina's hair fell out. To avoid embarrassment, she wore a wig.
I don't really like this suggestion as it is uninteresting though grammatical and easy to read.
2. I would rearrange the last part of the original sentence.
During the cancer treatment, all Gina's hair fell out, and she wore a wig to avoid embarrassment.
I like this suggestion better. It avoids a comma that breaks up the original sentence, does not lose the original meaning and is an interesting construction to read.