I was wondering if there is an expression to describe parents (usually newly-made) that are extremely happy with having children, with this happiness manifesting as "childish" behavior on their part when interacting with their offspring.
As an example consider someone with a 6-month-old baby that talks to the baby in a silly voice, makes cartoon faces and sounds, moves the baby around while simulating a train or aeroplane (with sound effects) and so on; at the same time this person does not hesitate to do these things in front of others -- possibly even complete strangers.
In Greek we call this person χαζομπαμπάς (father)/χαζομαμά (mother), a term that literally translates to "silly dad/mom" but does not carry negative connotations. At the very worst someone might call you that to imply that the parent is incapable of denying the child any request, but usually it simply means "this person is so happy that they don't pay any attention to appearances".
Is there something similar in English? I have already looked at this question, but it's not exactly what I 'm looking for because I read it as having to do with feelings, while I am specifically interested in how those feelings manifest as behavior.