How to replace existing ethernet wiring in the house?
Rip the walls open. Unless someone had the sense (they almost certainly did not) to run the network cables in conduit, that's generally the only practical method to replace wires thoughout a house. It's messy and tedious and involves a lot of cleanup.

Might be a lot less hassle to check for connector issues first - a 2007 house would almost certainly be using Cat5e (you can check the writing on the outside of the cable jacket if you have a few feet exposed anywhere), and Cat5e is gigabit cable - Cat6 has no practical advantage over it for Gigabit ethernet, which was designed to run on Cat5e cabling. Sure, it's higher frequency rated - but actual Gigabit equipment does not run at those higher frequencies, so that's pointless. Pure sales gimmick. 

[You'll naturally find many websites trying to convince you that Cat 6 is a magical secret sauce - Cat 6 is an orphan - it was supposed to run 10 Gig, but it does not - Cat6A does over short distances - so the cable companies were left with this cable they had gotten set up to make, and they whipped their sales forces to sell it. I personally have over 1/2 mile of Cat5e running gigabit on switches that report error rates, and I see NO application for Cat 6 - period.] 

Disconnects are almost always bad connectors. Bad wiring either never connects (it's broken, or mice chewed through it) or it connects at a lower-than-rated speed (it was installed poorly and became derated.) Low-quality and/or poorly installed jacks and plugs are a far more common cause of most problems, and especially intermittent connections.

Treating it like a new install, rather than "replacing the wires" you could abandon them in place and run wire though the basement, then drill up into the first floor walls, aiming for the service boxes,  make one run to the attic (either by ripping open walls that are easily repaired or scheduled for a repaint anyway (or in desperate cases go outside the house - or look for a space to run with the plumbing stack) and down through the tops of walls aiming for service boxes on the second floor. Slightly less mess. 

But I'd suggest concentrating on the connectors first, and possibly renting a network cable analyzer or hiring someone who owns one and knows how to run it, as being a lot more likely to solve your problem without making a mess of your house. Depending on exactly what your switch is, you might find that it has a built-in cable test function and/or error reporting - fiber optic switches circa 2007 were not the bottom of the line, in most cases.