Welcome from NJ.
I, too, would finish the cross stitching and quilting before trying to clean. You might be surprised to see how easily it cleans up! When you are at that point, I would soak in the bathtub with Dawn dish detergent before trying anything else. Some of the older dyes in the embroidery floss are/were not as stable as most of those today. I've had some issues in the past with some embroidery floss (on traditional cross stitching) but a couple of gentle washings with the dish detergent has worked wonders.
What I've also done with regular cross stitching after washing is a gentle squeeze to get out the bulk of the water but then I roll the piece in towels to get out as much water as possible and then lay flat to dry. This way if you need to do additional washings, you've not 'set' any runs or stains.
Good luck. It will be a much cherished quilt regardless.