While the above advice is valid and helpful for OTA TV service, the OP's opening line and later reference to verizon tech support makes me think this may not be the case here. In TLS's defense, I will say that the thread's subject is misleading. DTV generally refers to OTA TV, not a cable (and FIOS does qualify as a cable) passed signal.
first off, that 75 - 100 feet is a significant length for a cable signal. Funky connections are a possibility but so is a weak signal. The down side is I don't think the service provider makes any guarantees for service on lines other than those they ran.
Thge best (only?) way to ascertain if the length/signal strength is a problem is to lug that big mutha TV set close to the junction box and plug it in with the same cable you use for the working DVR. If it is a signal strength/cable problem, it should work. If not, well...
AB, how did you acquire your DVR? Was it from Verizon? Some service providers pass digital signals gratis, some don't. Back in the analog days, simply splitting a signal would work unless a particular channel was scrambled.
Nowadays, many digital devices have an internal address that's used by the service provider to turn on and off services from a central office depending on the subscriber's plan. They must senf a signal to that particular address to make it work. If it's not one of theirs (and/or you aren't paying for it), they most likely won't. A few years ago, they had something called "cable cards" that plugged into devices that were supposed to do, but there were problems.
If it is frrom Verizon and your billing reflects it, then they DO have to make it work. If you got it somewhere else, they have no interest in making it work. From what you say, since it only picks up remnants of analog broadcasts, I'd say all digital service is turned off.
good luck and let us know what happens.