You can still do it, in theory. They are just in the drywall. What you want to do, I theorize, is open the jacks up off the walls with a flat head screwdriver (most likely), and snip the cables at both ends. Tie a spool of Cat6 cable to the end of the one, and pull it through until you have pulled the old garbage cable and left a trail of Cat6 running from point to point. Then cut and crimp them to all new wall jacks. Get yourself a decent network switch, and configure the ports and patch from the wall jack over to the ports, and you’ve got a good home network. In theory - I don’t know if this would work, but I think it would.
My own house was wired similarly - but it was run like garbage, with no thought to running unshielded network cables across power and the interference coupled with super cheap patch paneling at the termination point just completely useless, so I just run cable through the ceiling where I want physical.