First off, don't feel bad. Most of us struggle with certain parts of the hunt. I have been hunting my entire life and have killed lots of animals. I still tear up and get choked with every single kill. Depending on which of my buddies are with me, there might be tears as well. Don't let the macho mentality scare you away. We are taking the life of real animals that live free and in the wild, that is a tough thing to grasp at times and a something everyone makes peace with in their own way. I am not particularly squeamish of blood and such, but there are definitely parts that get a bit gruesome. For me, once the animal is dead it takes on a completely different feel. I give my respect to life I took when I first walk up to it, but once the work starts its just meat. For me its not much different than walking through the grocery store and looking at all the different cuts of meat. At that point its just a carcass and food that will feed my family, it feels way different.
All that said, there are things that you can do to help out. Look at the gutless method. I process all my elk that way, even if I am closer to my truck. It just seems quicker and easier to me. And that way your not all up in the guts and stuff. That might be a good start. I completely understand the need to process your animal if you kill it too. However I would recommend having a buddy there to help that has done it before. First, I hunt solo most of the time. Processing an elk by yourself is tough work. It's a lot better with someone there to give a hand. Second, I never had anyone really teach me how. I learned through experience when I was young, and youtube videos later on. If there is someone that knows what they are doing they can help guide you through it, even if your doing all the work.
Your giving this thought upfront and realizing where your limitation's might be before your in the situation, that's all really good. it sounds like your a respectful outdoors-man already, welcome to the family!