That's a bit strange. There is a generic ACA rule of thumb that you have to apply for coverage before the middle of the month prior in order to have it effective the first of the following month. But in your case you applied around 5/9, which seems like it would be soon enough to start 6/1.
Suggestions:
1. Call the WA state exchange, explain the situation, and see if they can get it to start 6/1.
2. Go bare for half a month and just be super careful. I know most people don't like this idea, but I include it for sake of completeness.
3. Go buy a short term policy that will cover your family from ~6/15 to 7/1. You can probably get something like this directly from an insurance company...short term policies don't have to be bought through the exchange and don't have to be ACA-compliant, but for 2 weeks you probably don't care.