I think it's "the meaning that we were born", not just two of us but "we" in the general meaning.
痛みさえも抱えながら is a single clause, and both interpretations, "I will go to find the meaning while holding the pain" and "I will go to get new sights while holding the pain", are possible.
This might be a good example of the word order change, too. If it's この世界に生まれたその意味を君と見つけに行こう痛みさえも without 抱えながら, it would be barely possible to interpret 痛みさえも as the object of 見つけに行こう. Although I would use (たとえ)痛みでも/痛みだとしても for the case, instead of 痛みさえも. ("Even if it is painful" should be 痛くても.) But, actually, the verb 抱える usually requires the object to be stated. So, we can't put another noun/noun phrase between 痛みさえも and 抱えながら, as same as for the case of 逃げ場所探して. For instance, the order 痛みさえも新しい景色抱えながら has completely different meaning to the original; "while even the pain is holding new sights".