Those who are framing the issue in terms of "dealing with rejection" are completely missing the point.
If I send out 100 applications, get 99 PFOs, and 1 job; then I would be ecstatic. As a job market candidate, what I'm most concerned about is not avoiding or minimizing rejection; rather, I'm most concerned about securing the opportunity to do professionally the teaching and research I've been trained to do for the past ten years of my life. That's completely different from the experience of having a paper rejected.
There are candidates who had the misfortune of going on the market for the first time back when things blew up in 2008. These are people who know how to persevere. Every year they publish or otherwise further their research; they design and teach new courses to improve their teaching portfolios; they apply for jobs; sometimes they get interviews, sometimes they don't. These are people who are used to getting knocked down, and they know how to get back up. Telling those candidates that they "need to learn how to deal with rejection" is incredibly fucked-up and condescending.
This resonated with me because I've been on the market since before things blew up in '08. I've consequently been rejected hundreds of times. Almost all the jobs I've ever applied for have declined to interview me; almost all the search committees who did interview me took a pass. Including all the tenure-track ones. In that time I've had a lot of chances to really hone my skills. I don't want to brag, but I have gotten to be a pretty awesome rejection-dealer. I go out every year, get no offers, and I'm ready to try again next time. Dealing with rejection is not my problem.*
And so I feel qualified to say that one thing that I really do find helpful in dealing with rejection is this: don't make things harder on yourself than they need to be. Don't make rejection harder to deal with than it already is. If staying away from the APA makes it easier (or possible) for you to be ready to attack the market again in the spring, then you should stay away without feeling like the desire to do so is evidence of some character flaw. Do whatever you need to do.
Of course, maybe staying away from the APA doesn't help you. Maybe you get energized by the fog of misery, stress, and failure. Maybe it just doesn't matter to you. That's possible. And if that describes you, by all means feel free to ignore my advice. Do what you want. But I find the APA-with-no-interview experience to be intensely miserable, and I don't see the point in putting yourself through an intensely miserable experience just to prove that you can handle it.
*The rejections themselves are.