I've never heard about a Lime Diet, but that sounds like a fad diet. Most fad diets are fairly bunk (usually depriving one of various food groups, only eating one food, eating at certain times of day, and never mentioning exercise or overall wellness). Note: I'm not bashing the Atkins diet; it may be effective for a while, but eventually you have to get your carbs. The waste products from forcing your body to burn fat instead of both fat and carbs are not entirely healthy for the long run.
Anyway, back to weight loss. I'm studying athletic training in school (which is sports medicine), and weight loss and control is one of the many things I have to learn. Plus I just had a class on this very topic!
Two classic things are excellent for weight loss; exercise and a healthy diet. Reducing your calorie intake is good, and it's a start. But don't starve yourself. As has been mentioned, then your body starts to think it needs to conserve energy and your metabolism slows down. In order to burn calories, you would like a faster metabolism. So don't skip any meals.
To cut down on your calorie intake, start simply. Skip the second helping, or just eat slightly smaller portions. If you can't kick the junk food habit entirely, then try to compromise. Instead of having a small bag of chips, have half the chips and a few carrot sticks. The next day have the rest of the chips and some more carrot sticks. Take more veggies as your meal and take the smaller steak. That kind of thing.
Try to increase your activity level (that increases metabolism). The 20-30 minutes, three times a week is a good place to start. Make sure you warm up first (say, five minutes on an exercise bike or light jogging, enough to break a sweat), then stretch out really well. Toe touches, hurdler's stretch, Achillies stretch, etc. Do 3-4 repetitions of each stretch and hold them for 15-20 seconds each. Then do your 20-30 minutes of exercise.
Make sure you're doing an exercise that's fun for you. If jogging or biking just drives you bonkers with boredom, see if your local health club has some alternatives (aerobics, water exercise, or anything else). Be sure your exercise does push you somewhat. You can't get any stronger/faster etc. unless you're doing more than you normally do. Sweat a little.
On the flip side, don't gray-out, go until you collapse, or otherwise try to overdo it. That's counterproductive to having a good exercise session.
After your exercise, cool down for 5-10 minutes the same way you warmed up,
and stretch out again too. It'll help with post-exercise soreness.
The days you don't do more intense exercise, try to remain more active. Take a nice brisk walk around a block or two. Do some yardwork or cleaning. Park farther away from the store entrance. The whole point is to remain more active overall. Try to get someone to exercise with you, and you'll both remain on task more easily.
In regards to the sleep apnea, my father has that rather badly. He sleeps with a sleep machine that goes into his nose and mouth to basically prevent the snoring. He was, pre-machine, essentially waking up dozens of times a night and never getting really deep sleep. Hence poor metabolism, irritibility, etc.
Anyway, I hope that does help. Good luck!