RG-59 old ass thin cable not worthy of a digital signal. It will tile, it will cause loss over shorter distances, it can cause data drops to cable modem.
RG-6 is the thicker upgraded coax. The providers responsibility stops at the end of their line outside the house. They can get it working inside for you, but that's all your lines. You could pay a 1 time fee and have them replace it, or see if there is a maintenance plan (like insurance for cable) couple bucks a month and they take care of all your inside lines at no charge.