"signing the blue"

"But when the wedding season runs out, and the once-jingling
tambourine falls silent, those academic diplomas can keep a girl from
singing the blues."
what does it mean?

the blues is a genre of music that arose out of Black American folksongs (spirituals), characterized by melancholy sounds and lyrics and blue notes.

the blues is also a state of being depressed, i.e. feeling blue

So when someone's singing the blues, they could literally being singing the blues or they could be crying about some recent misfortune that befell them.

correction: being -> be

My guesstimate:

A woman is married and in this process there is a wedding ceremony
and other associated activities. Music is part of these activities. This is
all very exciting whether she realizes it or not. After all this excitement is over, a big let down (in a number of cases depression can result) often occurs. Going to work is an antidote to this let down/depression. The woman's chances of getting a job and of getting are more interesting job are enhanced if she has academic qualifications.

The last sentence should read:

The woman's chances of getting a job and of getting a more interesting job are enhanced if she has academic qualifications.