getting your "stuff" There
As I don't know what airline you are traveling on, and can only give you some general ideas. However, I am a retired airline pilot.
You will be allowed a very specific size on your carry on!!! Often airlines WILL NOT BUDGE!!!! A whole bunch of flies could possibly cause some problems, but I don't think fishing line is an issue any more. I assume you will want to take your rods on board too. Some airlines are very fisherman friendly(Alaska,Southwest), others are not. Get on the website and read the rules. Then, call to be sure you understood them correctly. Get the initials of the person you talk to and their position and phone #, and keep it with you.
That said, here is what I would do. Put all my fishing gear to include waders and boots in a sturdy box and fed-ex or UPS it .I would have it arrive a week early. VERY SELDOM does fed-ex or UPS lose anything, but in case you are the "guy in the barrell" get it there early. Thus the only thing you have to get on board are your rods!! Any questions, PM me.