For the UC’s and CSU’s, you have to meet the writing requirements by time of transfer so if they are not transferable, then you are out of luck. The best thing you can do is read the course descriptions of the needed writing requirements and match them to your private university courses.
You can use assist.org to see which CC courses transfer to the UC’s and CSU’s and find comparable courses at your current university.
I would find another counselor/advisor if they are unwilling to help since you are paying college tuition for their expertise.
Which Private University are you currently attending?