I have no written rental agreement. Do I still have to go through the eviction process? Do the new laws apply to me?

It makes no difference whether the agreement was reduced to writing, or was only an oral agreement. If someone is paying you (or is supposed to pay you) to live in your property, you are a landlord and they are a tenant. All the eviction and landlord-tenant laws apply.

Ironically, it some situations it may be faster to evict a tenant than someone who lives in your property without an obligation to pay.