how do I get ethernet without running cables through my walls?

There are some adapters that use existing power circuit wiring but in all honesty they are crap. If you have access to your attic they all you have to do is get up there and run some cable yourself. If that if not possible just buy some white data cable and some cable staples to run your cable along the wall and the ceiling to get it into the room you need. No other real option to get the job done. You can always use wireless but that is just never as good as cable.

You can get Powerline adapters which create an ethernet connection between two wall outlets by utilizing existing electrical wiring but I'm not sure how well it would work for applications that are more demanding. I have a pair of Powerline adapters set up that I am using for a network printer and it has worked well for that, but that's a low bandwidth application. I haven't tried it with a computer.

Another way you could go would be to get a WiFi adapter for the PC if that's an option and it doesn't have it. A strong WiFi 5/6 network with good signal strength may very well work better than a Powerline solution.

edit: Or alternatively use a WiFi range extender that also has an ethernet port. In this usage case it's pretty much the same thing as using a WiFi adapter on the PC in question, though an added benefit would be a boost to the WiFi signal for other devices in the vicinity.

Get another wifi router to your room, then connect to the network and plug an ethernet between the router and your PC. Other than that, you will need the cables, even if they don't go through walls. And don't trust those "powerline" things, those are terrible!

Going through a similar situation right now.  I've tried powerline adapters and wifi mesh systems with an ethernet cable connected to the satellite, both of which were kind of spotty for the work from home set up I've got.  I threw a PCIe wifi card into the desktop and IT called me to let me know I've got unauthorized hardware in my system lmao I'm running ethernet through my attic this weekend.  Not sure what kind of flooring you've got but if you have carpet, you can run flat ethernet under it.  Did that at my old place.

Google: AC + AP

saw a tech tips video where could use an old coax run if you have a old cable tv wire in wall

Or better yet just a WiFi range extender that has an ethernet port. A second router seems unnecessarily redundant and convoluted.

I don't see the sense in having to manage a second router just to expand connectivity in this case. And now imagine if for example you wanted to do something like host a game or a voice chat server and needed to enable the requisite port forwarding to get it to work properly. Except now you are talking about going through two nested hardware firewalls instead of just one. Yikes, sounds like a hassle to me! It's better to keep things simpler in my opinion.

I took this option myself three years ago and it works great.

The age old issue. Wireless is awesome and all, but you can't beat wired speeds. Running lines through your attic space is the way to go, if it's feasible.

Using WIFI would be easiest, and can be more that fast enough today!   But you can easily buy either pre-terminated Ethernet Cable in standard lengths, OR buy some ethernet cable, connectors, and a connector tool to make custom length cables.  Then you can run them through walls, floors, or an attic from your router to the your computer.