I would be pretty annoyed too! I've lived in Washington DC for the past year or so and thankfully, DH doesn't work for the government - hence for us, life has carried on pretty much as normal. The local library is open, as is DD's nursery, as is the local hospital (private, obviously) and somewhat surprisingly, our rubbish was collected this morning. The postal service is still functioning, and so too are the emergency response teams - a good thing too, given the shootings this afternoon.
However, having just returned from the local park, it's a different story. One of my friends is their family's only breadwinner (works for a government department). She is in work for the next couple of weeks, but then her funding runs out and she literally won't be able to pay their rent. Another friend is a federal employee and has been told not to go into work and isn't being paid. She told me they can survive financially until the end of November. Yet another friend is working but for no pay - now that is a tough deal! The 'shutdown' is affecting many working families very badly indeed.
A lot of the US media coverage can make the smallest thing sound like a huge drama, so there's been so much hype around this, you may find sights and so on aren't quite as restricted. If you're coming to DC at all, a lot of private museums are still open and there's still lots to do - though the Smithsonian museums being closed would be a real downer.
Not quite sure what I'm trying to say. I would be gutted in your position, absolutely gutted, but there are some people here who are going to lose their homes and so on if the shutdown continues - so all the more serious. Equally, the US has so much that is run privately, and you can pay to do, that you will still be able to have a great holiday if you want to.