1. Hard sided case with your own lock, not a TSA lock. Ammo is ok to be in the same case. Leave the ammo in factory boxes, or other closeable container.
2. Check in at the counter, not on line. EDIT: you can "Check In" to claim your seats online ahead of time, but go to the counter to check luggage.
3. When you get to the counter, tell the person "I need to check a firearm, can you help?"
4. Follow their instructions.
5. Be prepared to open the case for TSA to inspect. This may or may not involve TSA handling your gun or asking you to verify it's empty or most likely involve a bunch of awkward staring as the TSA person pretends to know what's up.
I've traveled often with firearms and the process varies depending on the airport, mood of the counter person and general randomness. But, I've never actually had a problem if I just followed along with what the counter person wanted to do.
You can ask what the protocol is going to be on the receiving end, but I wouldn't necessarily expect Charlotte to know what mood Topeka is in on any given day. Your shotgun may come out on the same carousel as the regular luggage, it might be sitting off to the side in an oversize area, or you may have to claim it from the luggage office. Again, just be prepared to go with the flow.
EDIT: Don't be "that guy" that waives a printout of the TSA or Airline regulations around telling the TSA what they're "supposed" to do. These guys are .gov employees...they don't care. Just go along to get along and you'll be fine.
Edit to my Edits: I've started placing a bluetooth tracker (
https://www.thetileapp.com/en-us/) in the case. It helps me locate the case at the destination city if it doesn't come out with the normal luggage.