Just to do my due diligence and mark this thread as "solved":
Yes, the Asus Sabertooth 990FX R2 can most certainly handle an FX-9590 chip with no problems whatsoever.
However, upgrading from an FX-8350 to a 9590 is not recommended IMO, since an overclocked 8350 can match or surpass the 9590.
Ultimately, my solution was to upgrade to Ryzen. And ever since then, I haven't had any unexpected troubles with FPS.
Although, I think I should answer the question I originally asked, in case anybody else encounters the same problem.
Based on what I have read (and experienced, I suppose), modern games often don't take full advantage of FX chips. So even though there is quite a bit of CPU power available, the games just won't use it for whatever reason, leading to significant bottlenecks.
Now mind you, the FX-8350 (with my experience) is absolutely capable of running most games fairly well...depending on your standards and budget. I'd still recommend it for those who need a budget PC that doesn't completely suck at gaming. But for those who are wondering why their FX chip isn't performing as well as expected, that's why.