Getting collections removed may be difficult since you’ve already paid them, but yes that would be a good start.
It may be a little longer before you generate a score. You being AU is fine for now, but eventually both of you may want your own cards and not just be AUs.
Any loans reporting on your reports?
You mentioned joining a CU recently; if they have a secured card you may want to look at that, or possibly Cap1 secured as it graduates like Discover.
It may be several months before either of you have a score. Stick to your plan of working on the collections and possibly pick up one other card. Take it slow in your building