Your account makes you appear to be a subtenant, but when you start talking about filing applications with "management" and also paying them a deposit it becomes possible that you have a landlord-tenant relationship with "management" instead of simply being a subtenant. Please clarify.
You and your landlord are bound by Oregon law to give proper notice. In your case, you have not indicated that you have a lease term. If you're a month-to-month tenant, during your first year of tenancy either you or the landlord can terminate on thirty days notice. You are obligated to pay rent through the end of the notice period, even if you move out early. If your landlord has given you less than thirty days notice, then compels you to leave at the end of the insufficient notice period, you gain a potential cause of action against your landlord.
The security deposit issue is made complicated by the involvement of "management". You need to tell us more about that arrangement and describe the details of the document(s) you signed in relation to your tenancy and the deposit.

