:-D Welcome to the board, Mrs BP!!! First, post, too.
As you can see, we are happy to help, but need specifics to help.
You called it a top, but said the dresdins were stitched with a running stitch. If by that, you mean a loose, long stich, to turn the edges, and to just to keep the dresdens on the background, then that was the original maker's temp stitching. Is the background a "wholecloth", all one piece? If so, then that explains why it isn't just blocks at this stage of the construction.
My g-ma, born 1880s, used running stitches on her Sunbonnet Sue's. Then blanket stitched them to background blocks, removing the running stitches.
It would be more typical that the dresdens would be running stitched onto blocks, rather than the whole top.
A picture would help us to guide you.
I would think you need to blanket stitch all the dresdens to the top, removing the running stitches as you work. Then proceed with preparing a backing and batting. Muslin would have been the backing for this kind of quilt in the 30's.
Happy Quilting!!!