Hi Scuba!
Welcome to 3FC!
There are basically two kinds of exercise -- cardio exercise that elevates and sustains your heart rate for a period of time and strength training that builds your muscles (also called resistance training, weightlifting etc - it's all the same thing). Cardio exercise is your main calorie burner while you're losing weight, while strength training builds and maintains your muscle mass. Muscles burn calories AND help you look tight and toned when you reach your goal, so both cardio and weights are essential parts of an exercise program.
The 'achey' muscle feeling that you describe is usually only experienced when you're working out with weights because that's when you're stressing and building your muscles. It's called DOMS - delayed onset muscle soreness - and it usually hits about 24 hours after a weight workout. But sometimes people will get an ache when they start exercising or use unfamiliar muscles for the first time - sounds like that's what may have happened to you when you started?
Yes, you are most definitely burning fat when you use the Gazelle! An achey or burning feeling is not a necessary part of cardio exercise. The Gazelle is cardio exercise, not strength training, and works by elevating your heart rate and burning calories. You say that your goal is purely cardio and that's exactly what the Gazelle is. An achey or burning feeling has nothing to do with how many calories you're burning -- a much better gauge of progress is how high you get your heart rate. It sounds like you're doing just fine.
In addition to the Gazelle, how about adding in some strength training? Since you're too broke to join a gym, can you work in any weightlifting exercise at home? Do you have any dumbbells or other exercise equipment?