You should be able to use outboard digtal tuner with their cable adapter. This will have an HDMI output as well as audio outputs.
If your TV is set to ATSC and it works then you need an over the air ATSC tuner. If the TV is set to QAM, which is digital cable, then you need a QAM tuner. All QAM tuners should have ATSC but not all ATSC will have QAM.
If you use a "digital converter box" meant for an older SD TV you won't get an HD picture.It won't have HDMI out.
If you use the correct box without the adapter you may not get all the channels available. A QAM tuner will only work with unencrypted channels. You can try it without any harm.