Good news. I think I may have a solution to the buzzing in the chime. At least it worked for me. Most chimes should have 3 terminals. One is common, in the middle. The other 2 are for the front door and the back door. Just bridge the front and back door terminals with a wire.
I have a picture and an explanation on my blog post at the end. Please let me know if this worked for you.
Installing the Nelly Security WiFi Video Doorbell
I have an alternative (but probably equally as effective) solution. I was a little nervous about just randomly sticking a wire in there without knowing more about what was going on, so I googled a little more about the problem. People have the same issue with Ring Pro, August, etc. August has a solution called the "August Chime Power Board" that is listed on their store for $0.00 with free shipping, so I ordered it. I figured I would get contacted by them to confirm my ownership of an August doorbell, but I got a shipping notification the next day. I got the power board today and it was a 2 minute install and the buzzing is completely gone and my RCA doorbell still works great. Before I installed it I took it apart, it's a very simple board. Here are some pictures:
Maybe someone who knows a little bit more about circuits can tell me what exactly is in this board. To me it just looks like a resistor and some sort of IC. I'm wondering what this does opposed to the solution that @theneedfull came up with.
I would encourage everyone to refrain from taking advantage of August's "free" solution. I feel kind of bad for even tipping everyone off to this. Hopefully we can come up with a simple DIY substitute, or someone can confirm why @theneedfull's solution works and why it's safe