'would forever' is correct and usable in written English.
It is commonly used to emphasize a lasting effect. For example, “Her influence on the children would forever shape them into kind and generous people.”.
I would forever stutter over my consonants – I dreaded k's in particular.
I was, and would forever be, a birth mother.
He feared he would forever be Dr. Sex.
She would forever wonder if she had taken his spot.
It would forever unfit him to be a slave".
This would forever taint Cerdà's legacy in the city.
Without action, he said, Britain would forever be doomed to paying the interest on its debts.
But it would forever bind his reputation to work that even he insists was "contaminated".
Other residents said that though Mr. Zimmerman was acquitted, his life would forever be haunted.
It was a collaboration that would forever alter scientists' understanding of learning and memory.
Watching them, it finally sank in that their names would forever, and inextricably, be linked.
