Sam25, I just looked at the specs of your current machine which is decent. However if you have the "itch" to upgrade, keeping most of your other components than the cpu,mb and ram I would suggest going with the 1600x.
I have built 2 Ryzen rigs so far. the 1800X in my sig below which is a High end and costly Ryzen that is water cooled. I'm waiting for the release of the Big Vega to replace my 2 RX480s in CF with a single high end AMD gpu.
I had ordered the 1800x ($499 US-not cheap) and wanted the Asus Crosshair VI Hero mb BUT availability of that mb was so limited, I ended up buying an inexpensive Asus Prime B350 Plus mb just to run the 1800x. I had some Gskill TridentZ DDR4-3200 CL16 ram that just wouldn't run faster so I also ended up buying expensive Gskill FlareX DDR4-3200 ram which does run at 3200 speed.
Once I bought my Asus Crosshair VI hero mb, I had another Ryzen mb (Asus Prime B350 Plus) and DDR4 ram (TridentZ CL16 DDR4-3200-16g) so I had the basis of another Ryzen build. When the 1600x released in the US for $249, I had the answer to upgrading my daughter's old AMD rig.
I used a Fractal Define S case but you have a better one with your Corsair. I had a crucial 240g ssd for OS and you have an excellent Samsung EVO 120g. I bought a 1TB WD blue for data. I bought an EVGA Gold 550W PSU and you have a Seasonic 750W -PLENTY. I used my Corsair H110i GT (overkill for a 1600x but it was available) and you have a stoudt Corsair H80 which is excellent.
I added a 1600x to the asus B350 Plus mb and have an excellent rig. Since my daughter does little gaming I used a Nvidia GTX 1050TI which was the best performing gpu in the price point. I might give her one of the RX480s if I buy a Big Vega. You already have an AMD RX480 so you are set.
The 1600x, without OCing, runs all 6 cores at 3.6 with up to 2 cores OC'd as high as 4 Ghz.
TRUST me. I run my 1800x OC'd on all 8 cores at 4 Ghz and that 1600x at stock is as smooth with PLENTY of cores.
I WILL NOT bad mouth the 7700k. It is an excellent cpu that will serve you well. However, for the $$$$, I would opt for a 1600x, and decent B350chipset AMD mb, especially since you do not want to OC. If you go that route be sure to CAREFULLY pick DDR4 ram that runs at higher speeds.