Wired Doorbell Ghost Rings

I just received a Ring Doorbell for my birthday and I installed it to an existing wired doorbell system. 2 days later, at around 3AM, I heard the doorbell ring...but, the Ring App didn't show any motion or send any alerts. Last night at around 6PM, our doorbell rang again, and like before, no alerts on the App (we were home and confirmed that no one was outside ringing our doorbell). 15m later, the same thing happened. So, I called Ring Support and after describing my setup (we have a Utilitech Digital Door Chime and 3 entry ways to our house, each with its own doorbell, and a 30VA Transformer), the tech indicated that I should replace the 30VA doorbell transformer with a 16-24VA, which I plan to do tomorrow. He said that in a wired system like I have, the battery in the Ring doorbell receives a trickle charge and that too much output may damage the battery over time. So for now, I turned the breaker off for my doorbells. However, he really didn't indicate that replacing the transformer will fix these "ghost rings" I am receiving. Again, the Ring Doorbell itself isn't "chiming" or sending alerts, it's just the Utilitech Digital Door Chime that is ringing. Could this be because I have the diode on backwards? Before securing the Ring doorbell to the installation plate, I verified that it along with my other 2 doorbells still worked and they did. I assumed at that point that I had the diode on correctly. Do you believe that the transformer may be causing this? Keep in mind that we had no ghost rings like that before I installed the Ring Doorbell and we built our house in 2012. I don't think I am buying that replacing the transformer is going to fix these false doorbells. If anyone in the community has any suggestions I welcome them. Thanks for your help!