I completely agree with this post. I smile at the flight attendant as I board, and ask for the extender. It's either handed to me right then, or they bring it to my seat a few minutes later.
I am very conscious of not spilling over onto a stranger. Also, a word of caution. Seats are different depending on the plane. I learned this the hard way. I'm fine in a 737 (Southwest), but I flew US Airways (CRJ, I think), and it was tight. I could fit, but felt I might be imposing on my neighbor. I ended up asking another passenger if she would switch with me, so that I could sit next to my co-worker (we were heading to a conference). We're good enough friends, that my co-worker completely understood. The person I switched with was in the middle seat, and she was happy to switch to my aisle seat. My co-worker moved to the middle seat, and I was able to sit on the aisle next to her. On my flight back, I upgraded to first class at my own expense ($90 more) to avoid the situation.
If at all possible you have to take the emotion out of it. You have no control over what others think and feel. You can either wallow in the shame, or recognize that you're doing the best you can. Food issues are not easy to control, and the more you beat yourself up, the less successful you'll be at it. People will always judge. If it's not your weight, it'll be your looks, or your laugh, or your accent, etc, etc etc.
Getting on that plane is a positive step. Going on vacation with your family will create happy memories to build upon.
Be proactive in your happiness. If going on this vacation will make you happy, do what you need to do make it happen. Get the extender or go ahead and buy the second seat. Do what you can to make it a stress-free situation. Don't let others (including your family) make you feel shameful or guilty for being proactive.
I've been fat for a long time. But it's only been the last few years that I've been able to recognize that feeling guilty does no good whatsoever. It doesn't make me work harder at losing weight, it in fact, hinders it. I'm most successful when I just focus on living my life to the fullest. The more I stay focused on the present and possible future, the more control I have over my diet.
Best wishes on you upcoming trip!!