As far as any left over pieces of the alarm goes, the only thing I've found was where they patched the original cable back to itself. I followed the alarm reset instructions with the connector to the ctm both plugged in and unplugged and still get the buzzing.

 

Sometimes when people cut wires and then use a spliced in wire for repairs, they use the wrong gauge wire and also use butt connectors to make the repairs. Doing that causes problems also, it causes a resistance effect on the butt connectors and the same for the wrong gauge wire. A proper repair for any cut wire is the exact gauge wire, solder, and heat shrink tubing over the repair. Your CTM, in my opinion, by the looks of it is in dire need of replacement. I know it is an expensive part to replace, but by the looks of the picture you posted, it should not look like it is melted in the corner like that.

 

Not sure how to post a question on these forums so I'll try here.
My 2006 ram 2500 door chime won't turn off with the door open and the key out. Any ideas?

 

https://dodgeforum.com/forum/3rd-gen-ram-tech-65/
Here is where your question should be asked. Scroll down the page to where you see "NEW THREAD" on the left side of the screen. Click on that and ask your question.