First off, I tried looking at the explanation here. However, the reasons for 我が家 don't seem to apply to 我が身.
Here is the excerpt that doesn't seem to apply from Jesse Good's post:
The above also applies to 我が母校、我が国、我が家族、我が妻、我が故郷、etc. They all imply that you have a deep relationship, you are proud of them, show affection, etc.
Obviously 我が身 is not stated there (possibly intentionally?). I'm unsure how it is possible to "have a deep relationship, be proud of, or show affection" to yourself.
The sentence that I read:
What is the purpose of 「我が身」 in this sentence? Is it possible for it to simply be substituted with 私自身? Also, what is implied by 「我が」 in 「我が身」 that is different than the 「我が」 in the examples from the link above?