You could have shin splints, it can be caused by muscles in the lower leg that are swollen from overuse, or flat feet because your feet are not given the correct support (you said you didn't have running shoes). You should check with your doctor to make sure you don't have stress fractures in your legs.
What you can do is ice your shins and generally let your body rest because it needs time to recover.. During this time, you should consider eating healthily if you haven't yet, and of course getting proper shoes for your feet. The shoes need to give you support in the arch (middle) of your foot.
You should modify your routine a little bit, until you've lost some of your weight. Jogging is a high impact exercise and you have 2 things against you right now. One of them is your weight; that's hard on your shins, ankles and feet to start with. Add to that, the fact that you don't have proper running shoes to cushion the impact (it does make a difference) and you could be in for a lot of trouble down the road. If you cause yourself to get shin splints or sprain an ankle, you could be forced to stop jogging and of course, you don't want that.
While I would never discourage anyone from exercising, I will most certainly discourage you from doing it in such a way as to injure yourself. We can rarely do all we want, right off the bat; we almost always have to work into it.
I'd recommend that you modify your routine for a while and instead of actually jogging do interval walking where you walk very fast for 60 seconds, then slow down for 90 seconds. Try this until you've conditioned your body and maybe lost a little bit of weight, then work gradually into the jogging and instead of doing it for 20 minutes, do the walk/jog for 5 minutes to start, with the remaining 15 minutes alternating slow and brisk walking. When you can do this and be pain free, then increase the walk/jog to 10 minutes with the remaining 10 minutes with alternating slow/brisk walking. Then go to 15 minutes walk/jog with 5 minutes of slow/brisk walking, then on to 20 minutes of walk/jog. Any time you end up with pain, you should take it easy until the pain is gone, then ease back into the exercise.
Keep in mind that you may have to try other types of exercise, as well, until you lose enough to reduce the impact.
You should also try some other weight bearing exercises, such as those using hand weights, crunches, squats, lunges, leg lifts, etc. These will build your muscles and muscles burn fat, which is what it's all about.
Check with your doctor about whether to use heat or ice on your shins. Some recommend one, some recommend the other and some recommend alternating.
Last, but not least... keep trying and be willing to modify your routine to whatever it takes ... once you start losing weight, you'll be able to do all the jogging you want. Go girl..