Can you stream media to your TV? If so, Netflix would be my first choice for classic TV shows. You can also get shows and movies on DVD or BRD from Netflix, which isn't the case with any other service.
DirectTV is just a satellite service, the content probably won't be much different than what you're getting with your TWC service. Check your TWC channel options and see if there is a classic TV channel available or if you can upgrade to a package that includes one.
