I had not even had my beta yet -- I just had positive pee sticks. It was fine. I ended up having to tell a colleague (she knew I had been TTC for some time) because she was trying to change around the time we were flying from LAX to SFO. I finally blurted out I couldn't change the flight time since I needed to give myself a shot right in the middle of the new flight time she wanted. I handled doing the shot in the airport bathroom -- no way on a plane!!
Good luck -- and hope you are indeed traveling when you are very early pregnant.