There are several ways to evade NAT. Your router may or may not block those, on a case by case decision. You must know really well how it will behave to have any confidence that you are secure.
Or, in other words, NAT isn’t a security feature because it’s not built to provide security. It may be intentionally broken so your home network works better, and it may have obvious security flaws that nobody looked after. For network security, you should use something intended to give you network security.
To check if your problem is caused by excessive memory usage requiring constant swapping. If it is, turning swap off will make some process be killed instead of slowing the computer down.