I don't know the size of your grey tank, but ours is about 10 gallons, and so boondocking is a challenge. We could go 4-5 days, being very careful.
If you must go without reservations, go anyway, stay in National Forest sites, and think of it as a challenge.
We spent about 3 weeks in 2013 roaming the Rocky Mountains, staying mostly in National Forest sites, and found a couple of things which stretched our holding tank capacity a bit. It was a wonderful time, in beautiful, forested sites.
I found a collapsible bin online, 5 gallons, I think, which I believe Camping World may now sell, that we stood in to take navy showers. Most of the water ended up in the bin, which we then dumped a ways away from our campsite.
You can take a decent shower in a couple of gallons of water. Sponge baths are also better than nothing, but a person like me who sweats
needs a proper shower every other day or so, even a spartan one.
The cooler temps at elevations helped me stretch out time between shampoos, too.
We also bought a couple of 2 1/2 gallon solar showers, which worked very well, and which we used ouside to wash our hair. These take up very little space when empty. When you are filling your fresh water tank, these can be filled as well and stored in your shower. I don't recall the brand, but if you google "best solar shower", they will come up. You can shampoo outside, into the collapsible bin.
That said, some high-bear-traffic areas in the National Forest sites specifically prohibit any grey water on the ground. Some of these have dump sites for dishwater/bath water.
Lots of paper plates and french fry bins for simple meals cuts down on dishes, too.
It could be a great adventure!