Upgrade suggestions, Ryzen 3700X/3800X vs 5600X?

[InfiniteNoise]

So my Ryzen 5 1600 (already overclocked, 3.8GHz, can't go any higher) is starting to show its age and I'm looking for an upgrade but I'm on a budget, so CPU is the only thing I can replace, I've looked around and it came down to 3700X/3800X/5600X since these are the best I can get that the budget allows for, my motherboard (Gigabyte B450 I AORUS PRO WIFI) is on latest BIOS, so it's pretty much a drop in replacement.

5600X seems like the best option for gaming, but I also do other things, including streaming, where 3700X/3800X seems to do a bit better.

About 3700X vs 3800X, there's only 20 quid difference between them, so I guess there's no reason to go for 3700X unless thermals is a concern with 3800X having higher TDP?

Other info that might help with the suggestions,

Case: Raijintek Metis Plus, airflow isn't amazing but I have the side panel off which helps
GPU: RX 5600 XT

The RAM is left at default, so it's running at 2133MHz, can't overclock/use XMP at all or I just run into random freezes, so that's something to consider, I know Ryzen CPU likes fast RAM but for now I can only upgrade the CPU.

I play games at 1080p on a 165Hz monitor, though I have the max refresh rate locked to 120Hz.

 

[InfiniteNoise]

I considered 5700G but if I'm gonna go for that, I might as well go for 5800X and get a bit more performance, it's only £20 more, and the double amount of L3 cache on it is quite significant, but then I'd be going quite a bit over the budget, I set it at £250 and I'd rather not stretch it too much.

I did some more research, and based on my usage, looks like I'll be going for 3700X, just leave it at stock and let PBO/XFR do their jobs.

I'm in no rush to upgrade though, I can hold off for a bit longer and see if I can snag something for cheaper when Intel's Alder Lake CPUs are out later this year.

Edit: turns out I can make the budget bigger and I just stumbled upon a 5800X at a decent price, decided to snag it.