Thank you. Here is an update:
I have not gone in to editinng the .dtsx files, a thing of last resort and maybe nto even then. I have been exhausting every check and look into every possibility. I've deduced that the actual tool, in this case, VS 2013 Shell Integrated with SSDT is bad and is on the server and where it is used via RDP.
I've back checked with using VS15 from a PC to connect to the server where the files are, and can call them up and see the connectors. So I'm able to determine that the .dtsx files are ok. Also, I have opened the .dtsx file in VS13 on the server, and can search the file and find all the references and connection details in the file. I've tried getting at the work files may different ways. Even ran a repair on the VS13 and it didn't help.
Thankfully the packages running as jobs in the server are fine, they run, and I can go in a couple of different ways and drill in and see the connectors. I plan to import from here on an as needed basis as a point of another go-to back-up if need be.
Now I'm going to make another backup of the works files on the server. Leave VS13 in place. Install VS15 and SSDT, and proceed from there in hopes that it too will behave as my PC local install of VS15, as in, reads the .dtsx files and the connector data within.