Problems with draft website pages

Possibly yes, possibly no.

I would recommend setting them to private and published so that you can see links working.

Drafts cannot be selected as menu links, nor do they have permalinks until they’re published (only the shortlinks and only while logged in), so that is why they aren’t working.

If you’re afraid of taking down the site, create a backup, set up a “maintenance mode” to hide your site, and then do the switch and fix what you need to fix. Then remember to take the maintenance mode screen down when you’re done.