I know a lot of guys use cotton, but I go ahead and spend the $ on the ear plugs made for horses. Smart pak and Dover both usually carry them. I’m not going to say don’t worry at all about getting them in the ear too deep, but don’t worry much. You’ll see what I mean when you go to put the plugs in, it would be very hard to actually lose a plug in an ear. Some horses don’t mind them, others take a bit to get used to them and are fine, and some just hate them. If a few hours after you put them in your horse hasn’t forgotten about them (still shaking his head, etc) it’s best just to take them out, they aren’t worth a sore neck.
I feel your pain, I have an older horse at the farm who can’t stand the 4th of July. Every year used to be traumatic. Earplugs didn’t seem to help him. What works best for him is to be in the barn, with at least one fan in his stall and on ‘high’ (he’s in Texas, it’s always hot), and with a radio turned up loudly, I go with either classical music or oldies country station, no death metal, haha. He still gets a little agitated and nervous, but nowhere near as bad as he used to get.
Good luck! I’m seriously considering starting a “4th of July horse owners co-miseration” thread.