Well, the newer the chips are, the less they overclock (for 90% of us, AMD overclocking is PBO and go)... so don't expect gains there/make that a reason to buy.
I don't think you've mentioned a budget, just that it's constrained. How much is 'just enough money for a 5000 series processor'? There are several...
Consider this......you have a 3900X already. Getting anything less than a 5900X would be a downgrade in core count. Since you're gaming and, presumably, don't utilize the core/thread count of the 3900X (right??) the 5800X3D is a good option.... but only if you don't require the use of the additional cores and threads for other PC work. So...........
Also, can you sell your current PC (mobo/ram/processor) and add that to your pile of cash? Combined, maybe that can get you the new AMD platform (or Intel for that matter).