Adding someone to your tenancy means you both become joint tenants. It's not the same as passing your tenancy on to someone else. You need permission from your landlord to add someone as a joint tenant.
Yes, you can legally be on two leases at once in Florida.
The landlord must approve the person, based on whatever criteria he sets for tenants and/or “other occupants”. He must either give you a NEW lease, with both of you on it (if the new person is going to be a co-tenant) or issue an addendum to the lease authorizing the new person to be an “other occupant”.