Hello Lynn. Mulching is a great way to retain moisture in your bed. by mulching the bed you will lesson evaporation of the surface and might create a haven for fungus and molds. It is possible that alot of mulching might change the contour of your garden and make water flow away from it. The answer to bad draining isnt mulching though (I am not an expert, people correct me if I am wrong!)
Double dug french raised beds do not necessarly need extra soil than what you have. I never have added soil; the technique though if done properly should add depth to your garden and break up the clay and base soil to provide better drainage. What I do is dig a shovel deep row placing the soil on the side of the row, then take a pick axe and break up the ground another six to twelve inches at the bottom of that row leaving the loose clay and soil there. Then, next to that row dig the next line taking the soil from the row and placing it on the broken soil from the previous row. continue until you have done the area of your beds AND WALKWAYS. Now, were your pathways are going to be shovel out that dirt and put it on your beds. I do a line with the shovel with the dirt going right and a line going left making a two shovel wide path for walk ways and drainage. Use a hard metal rake to create the proper shape of your raised beds ensuring you can reach the entire bed and have proper drainage. The added depth in the walkways, the added soil in your beds, and the broken base soil and clay deeper in your beds should create better drainage. Be wary of fungus and molds. Good luck. I hope to hear from you again. Madmark