Leadergorilla made some good suggestions.
Lee is a champion that is easy to pick up but be careful because he is hard to master. He is an early game champion so even if you haven't mastered him you can make some nice plays because he is so mobile and is flexible in builds. You can go full damage with him and get tanky later or go straight tank at the start. He also can build a sightstone allowing him to counter jungle, counter gank, and set up objectives nicely.
Kha'Zix is a bit more tricky then starting out Lee. His big thing is isolation damage and learning to pop in and out of stealth properly. Once you figure that out you have a few options. Depending on what you decide to evolve first and max he can be teamfight aoe or single target focused. His aoe just got nerfed and he may have more incoming though.
The thing about the jungle is it is very dependent on what your and the enemy's team comp is. Say you take Lee and the other team takes Vi. Mid and end game if you can't get off successful ganks Vi will outscale you hard. Other OP picks could be Yi and Rengar, but honestly I'd pick about 3 or 4 champions that have different strengths and power spikes. You'll be more successful then just picking OP champions because the jungle is so complex and there are a ton of factors at work. With the change of jungle items more junglers can be impactful if your know how to play them, e.g. a feral flare Udyr or Warrick. Too add, if you got frustrated as an ADC with the other lanes loosing I hate to tell you that it is going to be even worse as a jungler. Instead of them just losing it will now be your fault that they lost because as a jungler "you should be everywhere at once."
If your still interested in jungling I wish you the best. It is one of my favorite lanes. It is really a thankless position though. Before you make the transition though perhaps there are a few things you can do as an ADC to carry the game more. Jungle isn't the most reliable place to carry from. If your looking for more early game impact I'd switch to Mid because once you win your lane you can roam top or bottom and help them out. As for being an ADC, how do you do in lane? Are you winning it? If so perhaps notify your jungler and set up and do dragon. You could also suggest a lane swap or roam to mid and push down that turret while top is being ganked. The bot lane is the power house. If bot can roam because they are winning objectives can be taken since most likely both jungle and mid will be able to help you. Are you contributing to vision? If your winning your lane spending some change on some extra wards and placing them with your support isn't a big investment when you go back to lane. On another note, if you are having difficulties with peeling I'd do a few things. One record yourself and watch a match later and see if your positioning is off. I often get mad while playing adc that some one isn't peeling for me but after I calm down I realize that I could've positioned myself better. In champ select you could also do a few things. Going back to peeling you could ask if your support would pick someone you prefer for that. Maybe Jana or Thresh. You could also look at what champions you select. It seems you keep getting caught out in teamfights and the other lanes lose so you have trouble making an impact. Are you picking an imobile hyper carry or one that takes awhile to warm up like Cog'Maw or Jinx? If so maybe try something with an early or mid power spike and that is safer like Lucian, Corki, or Caitlyn.
I don't know your full experience so excuse I don't mean to be assuming that this is the case for you but I find that it works a lot better for me if I concentrate on my own play and improving and less on getting upset at my team for messing up and not winning. I've done that before and more oftent han not it causes my team to go on tilt instead. So instead of thinking "how can I carry this game no one else knows how to play?" maybe adjust slightly "How can I help out the other lanes so we can win this game together?."