I'm the same way- travel is stressful, but luckily, more often than not, the destination is worth it. I always breathe easier when I'm away from the city, so it's worth the anxiety. But I definitely am anxious when flying. I'm totally the "get to the airport 2 or so hours in advance to get through security and get to the gate" kind of person, even though there's maybe only one time I've actually flown and risked missing the flight. I traveled last week and was worried that the shutdown would impact TSA, but it really didn't; if anything, it was faster than normal (though that may just be the time of year). I'm also that person that, despite not owning any firearms or anything remotely dangerous beyond cutlery, almost always has that thought when going through security of "Is there a gun in my bag?" or "Will someone see the little plastic people in my suitcase and wanna embarrass me in front of everyone?" Quite the opposite, really- the one time I did have to get my bag checked because of a lightsaber, the TSA agents became kids again, it was quite heartwarming.
As a generally anxious person who dislikes both heights and going fast, being in the plane is a bit of an experience for me; taking off, landing, and any major bump of turbulence gets me nervous. Wasn't really that bad until a horrible bout of turbulence a couple years back. I know, statistically speaking, the chance of being
that flight are quite low, but that doesn't stop my mind from thinking it until we're safely on the ground. Luckily, I internalize it and don't make the flight miserable for others, but for me, I need to be constantly distracted or I get anxious as hell.
I know everyone has their preferred airline, but mine so far is Delta, if only because, on the off chance of the flight being oversold, they'll give ya basically a Visa gift card you can use anywhere, which really helps offset the cost of the flight or contribute to new toys.

Getting to where I'm going, especially if it's with family, I don't really like to give up my seat if it means less time with my loved ones, but on the way back, heck yeah. Got put up in a hotel one time by them and everything. Made me feel like Fergie- Glamorous.