Does your GPA for an degree carry over to a bachelors ...Speaking from public university background...?

No, your GPA from an associate's degree (or any other degree) does not directly carry over and become your GPA for a bachelor's degree at a public (or private) university.

While the credits you earned towards your associate's degree will usually transfer to your bachelor's program, the GPA is recalculated based on *all* the coursework completed towards the bachelor's degree. Your previous GPA is factored in only insofar as it determines which credits will transfer and how many more credits you need to complete the bachelor's. Your new GPA will be a fresh calculation using all the grades from both your associate's and bachelor's level coursework.