Judielaine, I found this article from NC State Extension
Morus Rubra which said under the Flower section...
"The Red mulberry has unisexual greenish flowers in small catkin-like spikes that appear in early spring with male and female flowers usually appearing on separate trees (dioecious). Trees with only male flowers obviously never bear fruit. Fertilized female flowers are followed by sweet blackberry-like edible fruits (to 1” long) that are reddish to dark purple in color. The fruits are sweet and juicy and may be eaten off the tree."
Based on that info, like you mentioned, if it was me I'd either plant a few more of the native red mulberries to ensure I had trees with both male and female flowers or plant a variety like the Illinois Everbearing.
Hope this helps!