where are you buying the phones?
its possible that they are being cloned.
a cloned phone means that somebody got the exact internal information from your phone and duplicated it. the part that tells the cell tower what phone it is and what company is carrying it and lets you make the call and allows the company to bill you for it. if they put that information in another phone than they can make calls off that phone which will be billed to your phone.
as to the internet it seems strange that the internet company would not know if you were connected or not.
if you go to this webstie
and tell me what the IP address is i should be able to tell you what carrier it is with. that might convince them. I supposed it is possible that somebody cross wired your cable service to run off of theirs but i really cant see why any body would want to do that. the opposite so that they could get free service from your bill is more likely.
that ip belongs to verizon. doesnt sound like it would be coming from a cable company
you would have to do the test from a computer in your house not from your phone or a library