Leave out わたし altogether. I think your first sentence without 私 is fine. There is a に vs. で question here, but I feel like they are both grammatical, but say slightly different things.
As for を vs. が, it's not the noun you need to look at -- they're determined by the verb. 交換する (こうかん) is a transitive verb, so its objects take を. が would mark the subject, i.e., the one doing the exchanging.