Most, if not all, cable TV and satellite wiring is RG-6 these days, the frequencies involved now are much higher and the old stuff has too much loss. Coax gradually degrades over the years anyway, gets lossier, it's probably time to pull some new stuff. If you've got a crawl space or an attic, it's not that hard to rewire things, electricians do it all the time without tearing out whole walls. A fishtape and a long auger bit go a long ways in that business.
On the O.P.s problem, I don't recall ever running RG-59 for 10B2, we had some skinny teflon stuff that we used. I don't think there's a
10B2 network card made anymore, haven't seen one for a long time, new. The newer network stuff uses CAT5 twisted pair cable, this is available in a variety of grades, some only good enough for baling wire. If you can install phone wire, you can install this stuff. If it's going to an outbuilding and you can't pull the old stuff out of the conduit to get the new stuff in, there's wireless networking available, has some security holes that need to be plugged, but available. If you use CAT5, get some decent ends and a good crimper, oddly enough, the best crimper I have for the plugs is also the cheapest, the most expensive one leaves open connections. CAT5 also needs a hub, switch or router to connect individual PCs to. With only two machines or devices, you can make up a cross-over cable, more than that and you need a hub and a place to put it, at least. They're cheap.Stan