Hi All,
I wanted to follow up and let you know that my problem appears resolved and here are the steps I took to fix it:
1) Installed two of the MFJ-915's as suggested. They did not resolve either issue with my headphones or my washing machine. However, they did not appear to cause any adverse issue. In fact, I remove the hand-made air choke I had made out of PVC and coax after reading that it was not perhaps a good "recipe" and may make matters worse. When I did and installed the two MFJ-915's I think my overall noise floor and reception improved. So... win there.
2) I purchased a couple Ferrite snap-on kits assembled and sold by DX Engineering. I clipped one on several winds of the washing machine power cable. This did not work. I clipped them on all the power cables in my bedroom "shack". This did not work. But I left them all there for good measure.
3) I then purchased a Furman power conditioner from Guitar Center and plugged everything into it... I know.. random idea, but I had a Guitar Center gift card from Christmas for $30 and figured it was a good purchase, since I really didn't want anything else.

. However... that didn't do anything to the two issues.
4) (and finally!) I looked at how others had clipped a ferrite bead around several winds of the IC-7300 power cable
right at where it exits the radio! This worked!. Headphones no longer projecting distorted audio and washing machine no longer freaking out when I transmit on 20 meters!
So I don't know if it was the cumulative effect, but... that's everything I did to get this problem resolved.
Wish I had magical eyes that could see RF and watch how it's flowing around my house when I transmit. Would make this a whole lot easier!
Thank you for the tips and advice!!
Steve
(KJ4DMO)