I know exactly what you are feeling! It has been quite a while, but I can remember with great detail the one plane trip that will forever be embedded in my memory.
I know Avery is only 3 months old, but you might want to ask your pediatrician if she is old enough or weighs enough to take a tiny bit of the purple, not pink, Benedryl...the one that has the decongestant in it. The decongestant will help her ears adjust to the pressure changes in the plane. I think you can give it to her every 4 to 6 or 6 to 8 hours, but since she is so little, you might give her less than the doctor says and stretch it out depending on your connecting flights.
If she cannot take the purple Benedryl, ask if she can take a tiny amount of the pink. At the very least, it will help her sleep consistently.
I probably should not be telling you to medicate your child! Please call your pediatrician and ask for suggestions. They may know of something that works besides Benedryl or cotton balls. I never tried cotton balls, so I do not know if that works or not. Since Avery is so young, she will sleep alot on the plane anyway as long as she is comfortable.
Be sure Avery has a bottle to drink on the plane b/c that helps with ear pressure adjustment. If you are BF and do not mind doing it in front of other people, start after take off when you can tell the cabin pressure is changing.
If you prefer to be alone when you BF, explain this to a flight attendant after you are seated. Perhaps once the plane is stable but the seatbelt sign is still on, they will allow you to go to the restroom, back row with a blanket or somewhere semi-private where you can BF...not the best condition but it's a plane!
Sometimes, certain airlines still let people traveling with children, the elderly, or the disabled, get on the plane before anyone else. If the airline no longer does this, still ask, but before take off.
If they will not work with you, as soon as the seatbelt sign is off, be the first one in the restroom if Avery is awake and looks like she is in pain or cranky.
If you have to wait for the restroom, try massaging Avery's head, face and body to relax her. Let her suck on a pacifier if she uses on. She just might go to sleep. I still do this for my girls sometimes and boy does it help relax them and change their mood!
Lastly, you may want to give Avery some Infant Tylenol drops to help with the pain.
I truly hope I have not offended you with my advice of medicating your 3 month old daughter. I remember when my daughters, who are now 8 and 10, were that age and I think I would have been appalled at someone who told me to give them unprescribed medication. I have learned since then that a decongestant helps with their ears adjusting when flying. I am only trying to help and offer suggestions, but Avery may be entirely too young to take anything. First and foremost, call your pediatrician and see what the office says.
I am looking forward to reading the other responses to your question. It seems like everyone in my family (except me!) and many of my friends are pregnant. I have forgotten almost everything about babies and need a crash course!
Good luck and have a wonderful trip! I hope you find the perfect solution for you and Avery!