Now I have several thousands more in various coins I have recently purchased sitting on the exchange where I bought them, and I am terrified of attempting to get them off the exchange and into wallets without getting stolen.
As I understand it, it is likely that there is malware on my machine deep in the BIOS or something which was capable of stealing the keys as they were generated.
I am sorely tempted to leave the coins in the exchange, since that has 2FA protection and, after all, it's the process of moving them that seems to be the biggest security risk, but everyone seems to agree that's a bad idea and I do hate the thought that a 3rd party is in control anyway. Can I please be advised on how I can safely get my coins into secure wallets? Thanks.
As the others said, you need to give us a few more details.
Since nobody raised the question till now, are we talking about BTC?
Then, the second crucial part, you said you've downloaded an offline generator.
Since this is the only thing you have done that is different from the previous time, assuming the machine was not used for anything else between those two points, what wallet generator have you downloaded?