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Yes, we are unmarried.  And yes, we have lived together in various stages:  We stayed for almost three weeks at her mother's house (maternal grandmother).  I just kept bringing a backpack of clothes over to the Maternal-Gma's house, so I never moved in. Then we lived separately in each of our own family's homes for a short time, and baby stayed with mom and maternal grandma.  Now we have been living at my mom's house (paternal grandmother) with the whole new family: our baby, my partner, and myself (plus p-grandma).  My partner still has well over two-thirds of her belongings at maternal grandmother's house, so she is settled but not fully moved in.
I apologize for the complexity.