2 Xbox One, 2 TVs, 1 audio system?



Hello everyone, completely new here to the forum and just seeking some advice. I'll start by saying I'm not heavily into surround sound and home theatre systems by any means and I don't plan to pour a bunch of $ into my set up to get what I need. I don't want the cheapest bottom dollar fix just something that will work until the equipment dies. Okay I'll apologize in advance for how long this is going to be..I appreciate the patience!
WIfe and I are using 2 xbox ones in the living room that are connected to 2 separate TVs. I need the TVs to share a single home theatre receiver that is connected to surround sound speakers. I'm not looking to get audio out of both xboxes at once but the idea is to be able to simply switch the input/channel/function on the receiver to whichever xbox we want sound out of at that time.
Some background on my equipment:
I had the same system set up before with a much older Sony receiver/dvd player that only had a single HDMI OUT so I had to use an Optical audio to RCA converter box that was fed by an Optical audio switch. (this also means that there were 2 more AC outlet requirements to power each that I want to get rid of) not to mention extra unnecessary cables etc.
The switch did have a remote to switch the optical channels each time (the remote broke so now I have to get up each time and so does my wife (who was recently pregnant but now a brand new mom as of 2/15/21 :) )
I thought I had solved my problem with the purchase of a Sony STR-DH590 receiver. 5 HDMI inputs, 1 HDMI output that supports ARC? 1 Single OPTICAL IN. Connected my xbox straight through the GAME input on the receiver and the HDMI output to my TV no problems there.
Then I connected her xbox HDMI directly to her TV and used an Optical cable from her xbox to the Optical IN on the Receiver (on this receiver you can assign which input that the optical in is assigned to) This seemed to work fine for less than a day until we switched back to her input channel from the GAME input on the receiver. This of course shut off my xbox signal through the receiver to the TV.

My questions are:
1. With this setup is there a way to keep my GAME input signal on and the video signal to my TV WHILE we are on her sound input channel on the receiver? Like changing any settings in the option menu for the receiver or any other inexpensive pieces of equipment?
2. If the above is a no, can I purchase a receiver that actually has 2x HDMI outs so that both xboxs go in their own HDMI inputs on the receiver and then from each HDMI out on the stereo to their respective TVs? With this method what would happen if I switch from one input to another, will it kill one of the signals like its doing now?
3. Can I purchase a receiver that has 2x Optical Audio In ports (one directly from each xbox) and then simply connecting from each xbox directly to TV with HDMI bypassing the stereo completely?
4. Is there just an all around receiver or set up that someone recommends that will do this without a lot of hassle or that's not crazy expensive?
Any help would be insanely appreciatedI I'm still able to return my receiver to best buy if I need to get something else. Thank you everyone for reading this long rant, I know how annoying/difficult it is to try and read something and then understand what that person is talking about or asking! Let me know if there's anything I can clear up or explain better. Like I said I don't know much about this stuff and I know I'm all over the place with the explanation! Thanks again!

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