You should tell your new employer what's going on - they'll almost certainly run a background check and see it. But absent some specific odd situation, they aren't going to care.

> if I was laid off "normally" tomorrow and got some severance package, that package still comes by way even if I got a new job on Monday?

Technically, this would be subject to whatever terms your former employer wrote into the severance terms. But again, absent a weird situation, they don't care, they're just trying to get rid of you.