But you can stop behind pretty much any piece of cover and switch safely.

You can just keep running and switch safely too.

Do you just switch while running around and pray you don't run into an enemy?

Yes, because running into an enemy exactly in the 300ms it takes to switch is insanely unlikely. I honestly can't recall it happening once, ever. And even when I'm halfway through weapon switching and and enemy appears, chances are you can still kill him quicker than he can kill you because he'll have a ~200-250 ms TTK weapon, and you just need to finish your weapon swap for 100-150 MS and then onehit him. Especially if you then juke by doing a sidejump or slide.

What do you do when you reload?

... Same thing. Just keep going normally and reload.

The only situations where I might stop for a second is if I'm literally about to round a corner or something. Although usually I'd just time it so that the swap/reload is finished as I'm rounding the corner, instead of approaching it, stopping, swap/reload and then keep going.

I honestly can't believe that someone would actually stop and go into cover every time they reload or switch weapons. In a CoD game.