I'd start with a new router that has plenty of inputs.
@zappas additional details on your set up need and what you are using the ports for would be helpful. If you want to keep your router then buy a multiport switch to plug into one of the ports. I had the same issue on my prior router and bought a $20 TP link switch that worked fine. It’s installed next to the router away from audio systems. I used it for non music function lines. Solar, office computer, gaming..the other port I connected my mesh system which had my music server and the mesh units connected to my audio systems. Now i have an upgraded modem/router with 4 additional ports. I use the ports for 1 switch, 2 music server, 3 mesh. Both worked/works for me but I prefer having my server plugged directly into the router now.
(1) Quality build and superior shielded Ethernet cable long enough single cable from the internet router to the network switch positioned close to your audio system ….say, within a metre away from the audio. equipment.
(2) You don’t need to go hog wild yet on an expensive “audiophile rated “ esoteric network switch expensive model ( think Silent Angel et al ). The latter custom units do work very well as audio performance upgrade for sure, but they are ka-Ching $$$$. I use a modest “ normal “ ( CISCO or peer grade home ….) 4-port Ethernet netwitk switch BUT ….and its a big “ but” ,with an upgraded 3rd party external power supply … go get an iFI POWER X.. Now the audio performance step-up results were immediate and a significant chunk the audiophile grade switch added audio performance, but at a fraction of the audiophile grade network switch price,
(3) Then you can plug in up to four shorter, - say 1m length - quality build Ethernet shielded Ethernet now cables into the four network switch ports
(4) the other end of the Ethernet cables are then connected to your Ethernet connected sources , ….player, NAS, streamer , DAC etc.
If you going to buy a switch they $20 make a ton of noise as well as your router .
this switch is the best value for all it offers and under $600. LHY SW-8
i spent a bit more from Linear Tube Audio they make a great Linear power supply up to 20 v and 8 amp capable , $750 for your router little green computer $299
get rid of those horrible $5 wall warts they destroy the fidelity especially if streaming.
I second all of the suggestions here, including the one about changing the router, but I use a basic Linksys switch that has 10 ports and have had no issues. I had looked at an uber$ Melco switch, since I use a Melco NAS/Player, but decided that the $1500 price tag wasn’t justifiable, especially since that is about 15 times more expensive than the router
If your Ethernet is somehow passing noise into your streaming device effecting the sound, you have a crappy streaming device with a bad network port that was not correctly designed to work in the environment as advertised ( Ethernet) Don't blame the network. The reasonable band-aid would be an optical isolator. The only reason to go to shielded CAT-5 would be if testing shows a lot or packet resends due to data integrity. 1000' runs in industrial environments for example. Or where electronic espionage needs to be protected. Sometimes required in medical installations. Both ends of the equipment then need to be compatible for the shielding and be prepared for sneaky ground loops. Good luck self terminating shielded yourself.
Audio bandwidth is trivial compared to Ethernet. Hubs work fine. OK, if someone was a high end multi-player gamer while someone else was downloading large data files between ports on the hub then it could get congested, Isolating ports in a switch vs a hub won't really help otherwise.
As far as sound, well remember there are two definitions of sound. Those that are variations in air pressures you ears sense. Real physical things in the world. Then there is what your BRAIN believes. Imagined sound. It may be real in the physical world, it may not. To you, it is real if you brain says so it even if it is not based on actual vibrations in the air. Drink whatever Kool-Aid you wish.
A 50-cent ferrite does wonders for those "horrible" switching wall warts. If you need more than that, again, your equipment is to blame.