The advice I was given on tranquilizing was don't do it. The reason being is that no one is going down into cargo during the flight to check on the animals and so if they have a bad reaction there will be no way of telling. (Even if someone was to go down, they're not allowed to open any of the cages, they're shut with zipper ties at the airport). Its a bit more scary for the pets, but far safer to keep them sober.
Delta will also allow pets on board, it just as an allocated number of spots, so you have to call them up and inform them. I have two cats (brothers) that I brought in last May with Delta. They had to go under because Delta has a requirement that they be able to stand up in their carry case AND that the carrying case fits under the seat infront of you. While I wanted my babies with me, I was pleasantly surprised to hear that rule (and didn't put up a fight since they obviously had the pets best interest in mind). I also took advantage of the fact that I could give them a bigger cage and put them together if I flew them in cargo. They had toys, drip water, a bed that is nicer than the one I slept on for two months (ie- the floor), and the staff even loved on them through the door while they were waiting to put them on. (Ok, the last part I only saw up until I passed the ticket counter, but it was reassuring none the less). Customs was super easy, my food was taken away but they weren't mean about it, just explained that because it didnt enter in a sealed and labeled bag there was no way they could prove what was in it. eh, not a huge deal. After that I was just told I had to keep them inside for 3 weeks and that I was free to go
They both live like kings down here now and were fully adjusted in about 3 hours, so apparently there was no lasting trauma