Great advice given.
One thing I will add, is that all too often, there will be someone at the airport or check-in, that will believe you are NOT allowed to do what you are doing.
Some years back, when we went hunting in Canada, I printed out the requirements/regulations from the airlines we were flying. We each had a printed copy of the airlines regulations. When someone said we could not do such&such we just calmly pulled out our typed up sheet and showed them where we could. It did end up saving us some major hassle with one employee (who because of our typed up sheet called a supervisor, who cleared it up).
Get the requirements and regulations for flying with gun/ammo, follow them to the tee, and have a typed up sheet with you, so that you can easily demonstrate that you did everything correctly.