I tend to have a "simplistic" view at times and prefer "simple" answers to complex questions. I know when I do get "simplistic" I get looks of, "is she crazy!?" and "she just doesn't understand"...yet, simple answers sometimes are the best.
I just wish her and I could stop being catty with each other and stop competing for his love.
So do it. You can only control your own actions and reactions. So do it. Don't be catty. Stop competing. It's the same thing when an arguement becomes a shouting match. All it takes is for one person to stop and wait. Person two may take some time to realize that they won't get anywhere by shouting, but they eventually do as long as you just sit and wait. No looks of "get over yourself" or rolling your eyes. If one person says something "catty" don't respond back in kind.
This can go for both of you. Realize when things are starting to head toward the "catty" and competing, and then stop. Take a deep breath. If you realize that you are being "catty" then accept it, say to yourself, at the least, "Dang it! I'm being catty again." Then take that breath and try again. Doing it outloud may help even more though as you both will start to see that you are able to control your reactions and willing to do so.
I did say I was into "simple" answers. This is true, the answer is simple, just do it...the practice of it may not be as simple. I wish you both, all three of you actually, the best of luck.
BTW: is DH on here as well? I can't keep track sometimes.