If you're in an urban area, try a set of rabbit ears or an indoor amplified antenna.
www.antennaweb.org can help you determine if an indoor antenna will help you.
As to the analog signals looking like grainy shit on your new set. It's because analog cable signals ARE grainy shit, you're just able to determine that fact due to the larger screen.
FWIW, consider an over the air antenna for your next move also. Cable companies often rob bandwidth from off the air HD signals in order to cram more channels into the limited bandwidth their coax can carry. It can be amazing at how much difference there can be between off air and digital cable!
www.antennaweb.org can help you determine if an indoor antenna will help you.
As to the analog signals looking like grainy shit on your new set. It's because analog cable signals ARE grainy shit, you're just able to determine that fact due to the larger screen.

FWIW, consider an over the air antenna for your next move also. Cable companies often rob bandwidth from off the air HD signals in order to cram more channels into the limited bandwidth their coax can carry. It can be amazing at how much difference there can be between off air and digital cable!