Need Help Choosing a New CPU for My Build

Hey folks! I hope this isn't a silly question, but I'm not very knowledgeable about PC building and reading articles just seems to make me more confused. I'm looking to replace or upgrade my current CPU, which is an AMD Ryzen 5 3600x. I have an AM4 socket motherboard.

I'm considering the AMD Ryzen 7 5700x because it seems like a better version of what I already have. I initially looked at the Intel Core i5-12400F, thinking it might be a good option since my friend has it, but I found out that it's not compatible with AM4.

My motherboard is relatively new, having been bought just about a year ago, so I'm not eager to invest in a new AM5 motherboard since I'm on a budget. I'm looking to keep my spending under $200, so both the Ryzen 7 and the Ryzen 5 should fit my budget. Are there any other good CPUs I should consider? What are your thoughts on my choices?

Also, do I need to buy a specific cooler for the CPU? Right now, I have a cheap fan and heatsink, but I noticed there are listings for CPUs that come with coolers too. Thanks for the help!