First off, you wont be burning calories like crazy, unless you eat nothing before you are working out, and are EXTREMELY lean. If you do it correctly, you can burn the calories you consumed from fat or carbs depending on duration of your exercise.
It might be a shock to you, but you burn about close to the 60 calories you mentioned in the form of (TEF) Thermic Effect of Food. If you ate a 600cal meal, 10% would be burned to chewing and digesting, etc. That's 60 calories.
So first off, you dont need to STOP the cardio. In fact you need to be doing the cardio. People think that when they get on machines and start burning calories, it is all in the form of fat, and this is WRONG. You dont start burning fat until a long time.
And if you are burning MUSCLE....you have SERIOUS problems with your diet, or you are 0% bodyfat (exaggerating) ok mr scientist man so he's not burning fat.. or muscle. when he uses the glycogen stores in his liver bloodstream and muscles after 20 mins, what's he burning then?
My suggestion, is that you EAT MORE. You dont list a diet in here for us to look at, but my guess is that you dont eat enough.
There is no way you will burn muscle on the elliptical or at cardio period unless you are starving yourself to death.
And the kids that warn you to stay away from cardio are the one YOU SHOULD NOT BE LISTENING TOO. They dont know their stuff.
If you only do 15-20min or so 3x per week or something, there is nothing wrong with that. Just dont be doing 3hrs a day or something silly. Then you are going to have to eat even more. But there is NO REASON you cant get the extra calories in you burn in about 20min of cardio.