So in the span of 5 days my refrigerator started leaking slowly and the cold cycle on my washing machine would not start.  Unfortunately I have hard wood in my kitchen so it took a day or so of dripping before I stepped on a water saturated wood plank to notice I had a problem.  After soaking up what I could I pulled the fridge out and found the source of the drip - the water valve.  So I turned the water off to the valve and did some GIS for common leakage issues.  Turns out that water valve problems are pretty common.  Thankfully I have a pretty common fridge so I was able to order the part from Amazon for about $40 less than any local shop.  
 
While that part was on order the clothes washer decided to not start on cold cycle.  Another quick search led me to an issue with the water valve in the washer.  There is an inlet for hot and one for cold water.  Depending on the cycle the machine will allow one or both valves to open, letting water fill the washer.  This was the cheapest replacement part, about $30 from a local parts store. 
 
Both parts were very easy to replace, both are now working as intended.  I'm very thankful for exploded views of appliances and the participation of internet users sharing their repairs.  I'm also pleased that I don't have to buy two new appliances...yet.  Now with the 100 degree weather we are having, I expect some sort of revolt from my HVAC.