Scam or not?

This morning, i got a call from (888)266-2278. They were asking if I am moving to a new place as they see a new customer starting a new xfinity internet service at my current address. This was correct so I told them I am also moving and would like to move the xfinity internet service to my new address starting next month. They offered me discount so that I can pay 50 per month for 500 Mbps with autopay. Even though, the bill sent to me via email says I will have to pay $60 per month, the caller explained that this is before the autopay discount so by the time I pay, it will show as $50. So I completed the process and they sent me a link for an approval. I got official emails from Xfinity about keeping the service to a new home. The caller told me to just bring the modem with me and set it up at my new home on my moving day. Then everything should work. So I am pretty sure this part was not a scam.

But later, I received another call from (520)200-2747 telling me that it is COMCAST xfinity. They knew that I changed my address this morning. Also, they told me they were calling me to avoid interruption in my account, I should cancel the service for the existing address and the new address (maybe I misunderstand this part. They said something like since I already paid for the next month at the current address, it should cause trouble or something. It was unclear what they were telling me to do). Then, they told me to call them back in May to start the service in the new home. So I told them if I cancel the service and called them back, will I be able to get the discount they promised. Then, they were saying things like plan package or something that I don't understand so I just hung up the phone. 

What is going on here? Is the second call a scam? Why should there be an interruption in my account? I think Xfinity should take care of it by themselves so I was very confused. Would you explain what should be done to avoid double charge or any scam?