I have a similar problem, and can provide specifics on my similar situation.
My situation is the cable company provided me a free digital adapter box (DAB) - non hi def - and for this room I have an older but still digital Panasonic TV (2005). The only type of digital adapter box the cable company has is coax in and coax out, and includes remote (similar to a regular cable box), but no other outputs (jerks).
Back in 2005, a friend and I were taking the TV off the wall to move it, and he (or me, can't remember) disconnected everything but the coax cable from the back. When we pulled it off the wall, the coax cable pulled the coax input out from the back of the TV. Everything works perfect on the TV, except the coax input. Have been using HDMI, but now want to shift this to a guest room in place of better model.
I either find a coax to HDMI (or even composite, or component) adapter, or I spend $10 more a month for ANOTHER standard def or hi-def cable box.