E., you can't be the only one in that boat. See if you can post a card at the PT office offering to trade sitting for another patient during PT appointments, also ask the receptionist and PT if they are aware of any other patients that have children that may be in the same situation you are (chances are if they do, they talk about it). If they know of someone, ask if they can pass your number along to that person and again offer to sit for one another during appointments.
Getting involved in a mom's group is a great way to network and find solutions for about every kind of problem. It's also fun for both you and the kids. I know of two in Kirkland, but they are all over, so I'm sure you could find one in your area.
I noticed a few people suggested going through your church, which is also a good idea. Some churces compile a directory of everyone's contact information and employ. If yours has such a thing, it would be easy to find either a sitter or other stay at home mom's that you could possible trade time with. Some church's also have mom's groups. In general though, you should be able to find some sort of support at you church.
I'm truly sorry your family isn't more supportive. They don't realize what they're missing out on. Although I also don't know your children and how difficult or easy they are to watch after. Personally, I think boys are easier to take care of than girls, but then again, I have seven nephews. I come from a family that truly believes in the idea "it takes a village." My sister and I have both taken nephew's for a week at a time to our siblings could go on vacation with their spouse. I have a friend that has 3 children (a set of tiwns and a four year old) and she hates the idea of imposing on her friends. I'm still trying to convince her that to me its a treat to watch her kids, not an imposition. You never know, you may have friends out there that feel the same, its just that you've never asked, so approach some friends you've never thought to ask before. There is no shame in needing help or asking for it. If I knew you, I'd offer to watch them once in a while.
Soon in both Kenmore and Kirkland, the outdoor concert season will begin and they're are alot of families with young children that attend. I know they have these in communities all over the Seattle area. You should find one near you, pack a blanket, snacks, bottles, diapers and give it a try. I'll bet you, your husband and the kids will all enjoy it. All genres of music are represented over the summer. No need to limit yourself to the doodlebops (oh, but I do love their latest single!- lol)
One last thing. On your accident. I obviously don't know the details or who was at fault. However, if you were not at fault and you have not settled yet - don't unless you are forced to by a court date. You will be surprised on how late injuries can begin manifesting from a car accident (speaking from personal experience here).
Good luck with everything. I know it can be frustrating, but it will work out. Just remember, there are a lot of women out there facing the same thing every day and they get through it. You will too, without having to let your health suffer for it.
L.