@Marjorielouise,
Because your mother is not forthcoming, I suggest you talk to her siblings if she has any, even inlaws if they were in the picture in 1968.
You can possibly try Ancestry.com (you may have to pay for good access) or the like, including sites like Classmates and FamilyLink. Look for the Mormon information as before the internet they used to keep records on all sorts of people in the US, including non-Mormons. I would also dig through Facebook as nearly everyone alive in the US is on it these days.
You can also check around at her high school or church if she was active then. I realize she would have graduated by then but that doesn't mean your bio father wasn't a classmate.
Consider police records, prisons, and death notices as well. There are privacy concerns with a lot of these so they might be a dead end. But think outside the box as any of these is possible.
Another option is a skip tracer or a private investigator; those people are most easily found via a lawyer. Be prepared to pay for this.
And also resign yourself to the fact that it might never happen.