If it is fleas or bedbugs, some of them could have taken up residence in your bed by now.
Can you post photos of the quilts?
Unless there are a lot of large holes where the batting could come out, I would risk washing the quilts individually in a top-loading washing machine. Just make ***certain*** to stop the machine before it agitates. Hand agitate instead (you can use a broomstick to push down on the quilt), and advance the machine to the spin cycle after both wash and rinse cycles. I would use extra cycles too. Any mild laundry detergent will be okay. I just wouldn't use too much of it at one time. I would maybe do two wash cycles with a little detergent in each one, and then rinse several times too.
To dry, lay on a large clean bedsheet -- outside is fine as long as it is on grass and in the shade (sunlight is hard on fabrics). Cover with another sheet to keep sunlight and bird droppings off the quilt while it dries.
It's possible that something other than critters irritated your legs, but the fact that it looks like bites is not good.