They should be considered valid for 6 months after the date on the background check.

If you want one quickly, get them done through a channeler. I used to use Accurate Biometrics but they recently started requiring you to send two sets of fingerprints instead of one, so I went with National Credit Reporting which only required one. You'll have to get the report sent to a relative in the US who can then send it to you. If you don't know anyone in the US who can do it then I don't know what to tell you. You should get it within 2 weeks, much better than the bad old days when you couldn't use a channeler and it would take 2 to 3 months to get an FBI report back.

If you really do need an apostille then look for a channeler who can do that, or a notary service that can do it.

Also I don't think you should have to get an apostille, I'm pretty sure that only Korean immigration really require it, and they don't require new background checks if you haven't been back in the US for more than a few weeks. An employer should not require an apostille themselves, and they shouldn't be asking for background checks anyway unless it's something immigration wants. Maybe look at other jobs that don't require it. Good luck.