I think they meant "diverging" not "divulging"

This sentence is incomplete: it doesn't make much sense on its own and it's difficult to guess exactly what it means. However, if I had to guess, the second half of the sentence could mean: "first, by acknowledging that you had divulged [something]..." where "divulged" means "made something known" or "made something public" or "brought something out into the open". The phrase "your divulging" can be expressed differently as "that you had divulged". For example: "He complained that you had divulged his secret" = "He complained about your divulging his secret" = "He complained about your having divulged his secret". Using "your divulging" or "your having divulged" in these contexts is grammatically correct (and I would say these things); however, it is much more common to hear people say "He complained about you divulging/ about you having divulged his secret", which many people think is clearer.

I guess they were acknowledging that they were revealing “divulging” private or secret information? It’s a bit hard to understand without more context. Maybe a few more sentences that came before this one.

Can be difficult when the savings are not just "there". Try to think, you were on the right track; first acknowledging your divulging. Think even if it is uncomfortable what is one small change or adjustment I can make to attempt to do this differently?

Think of a scenario like, you have an idea and you found out how to make that idea into reality and you know that you have to keep going until you finish. "Divulging" in your sentence means, "Keep going" or with known synonyms like "Discover" or "reveal". I hope that helps.