Adding the cory before the cycle was complete may have been too much stress on both fish. The added ammonia created more stressful conditions.
The temperature drop may have also contributed to the problem. Extreme quick changes are the problem, not the temp it dropped to. Are you sure you've got the tank at the right temp? I just did a little web surfing and found that cories like 72-79 degrees and bettas like 75-84. It may be too warm for the cory.
Also, the same site said cories are schooling fish and you should have a minimum of 3. Yours might just be lonley. I don't know anything about cories, so hopefully someone with more experience will come along and advise you.
If you are serious about keeping fish, you should buy your own test kit. Get a liquid kit that has two testers for the ammonia and nitrate. I have the API master liquid test kit that tests for pH, ammonia, nitrites, and nitrates. You need to monitor your own cycle and water conditions. As soon as you noticed something wrong with the fishies, the first thing to do would be to test the water, then post here for help -- noting the parameters.
I have a notebook and I write down the test results so I can keep track of any changes. You'll need to do that to monitor the cycle. I also keep track of when/how much water changes, medicines, when I add new fishies, any other major changes (like cleaning the filter), etc.
Keep us posted!