How to Fix Extreme Lag

The performance of an RX 560 is quite close to that of a GTX 1050, and in certain DX12/Vulcan games can even outperform a 1050 Ti, so replacing it with either of those would be pointless. The RX 560 should be a fine enough GPU for now, and if you're on a tight budget, it probably wouldn't be worth looking into a GPU upgrade until prices come back down.

The CPU is undoubtedly holding back performance more than anything. Unfortunately, I don't think there's much that's significantly faster on the AM2 platform, so you'll likely be looking at both a CPU and motherboard to get notably better gaming performance. And if you plan on getting a newer processor from the last couple years or so, both Intel's processor's and AMD's Ryzen require DDR4 RAM, which is also priced relatively high right now. Another option would be to pick up a used Intel i5 or i7 CPU from a few years back (2nd/3rd/4th gen), which can be faster and should work with DDR3 memory.

Another thing to check though is that the temperatures of the CPU and GPU are fine though, since an overheating component could cause it to throttle, exacerbating the problem. While your processor might not be great for gaming, I don't believe performance should normally get quite as low as you are describing.