Why would this be nem szédülök, rather than nincs szédülök?
Nincs replaces "nem van" (when you're talking about the absence of something, or something that doesn't exist). Other verbs like szedül just get nem in front of them when you negate them.
I see, so nincs would be used when talking about a thing not being there, but not used for verbs. I hope I understand that properly. Thank you!
That's right. Nincs is the negative form of the verb van (to be, to exist). And nincsenek is the negative of the plural vannak.
For every other verb, the negative form is simply nem plus that verb.