I use an 8 gallon actionpacker for extended trips like yours. You don't need anything more elaborate than that, and since it's not built into your rig, you can carry it to your camp shower area or take it into your tent.
My girlfriend uses almost the same setup as I do, except she likes to bring more tops and she carries a lot more shampoo and stuff like that.
I carry:
3 hand towels
3 sets of clothes - 3 shorts, 3 underwear, 3 t-shirts, 3 fishing shirts, 3 socks, hiking boots
3 sets of sleep clothes - 3 gym shorts, 3 tshirts and sandals
1 set of town clothes - 1 jeans, 1 dress shirt, 1 belt
If it's a trip with cold nights and hot days, I carry sweatpants instead of gym shorts for sleeping, and I carry two pairs of six pocket tactical pants in addition to the shorts. I might carry 3 additional and heavier socks if needed so that I have daytime socks and nighttime socks. I don't spend much time in towns but if you do, add another dress shirt or some Docker type pants. The gym shorts double as swim trunks. You don't need a big towel when you're camping. A hand towel is plenty.
All of the above, plus your toiletries, plus your sandals (in a big ziplock) will easily fit into an 8 gallon actionpacker. That's even including the winter stuff if you need it. In winter I also carry a hoodie and a jacket, but I never pack those - they stay out in the open all the time.
The idea with the 3 sets of clothes is that you basically have one set of clothes "in the wash" all the time. It takes a full day to dry stuff out, and you might have a rain day, thus the 3 sets instead of just 2.
Fold and roll everything up tight. roll up your underwear, roll up your shirts, roll up your pants. Rolls take up the least amount of space and have the fewest wrinkles and it's very organized. Even dress shirts can be rolled if you practice at it. They will come out with very few wrinkles. Look up "How to roll clothes" in a search engine and you'll find lots of info.
I usually camp in hot weather and I'm very active, so my daily routine is that I'll wear an outfit for only one day maximum. I always take a shower just before going to bed and then I put on clean sleep clothes (I don't like to sleep in campfire smelling clothes!). After my shower, I wash the sleep clothes from the night before, and whatever I wore today plus the dirty towel, then wring them all out and hang them to start drying before I go to bed.
My entire washing process takes about 5 minutes or less and you get used to doing it, it's not a big deal. I use a two bucket method to wash clothes. One 5 gallon bucket with a gamma lid, and one 5 gallon bucket with a bunch of small holes drilled into the bottom. Take your first 5 gallon bucket with a gamma lid, put in the clothes, soap and water, put on the lid and shake it up real good for a few minutes, dump the soapy water, add some fresh water for a rinse and shake. For the wringing part, lay down some sticks on the ground. Put the second bucket with holes in the bottom on top of the sticks. Put your clothes into the bucket that has holes in the bottom. Next, put the first bucket with the gamma lid on top of the second bucket and sit on it. The first bucket will press all the clothes down and the water comes out of the bottom of the second bucket. That's it, you're done - the clothes are completely wrung out and clean. Hang the clothes on the clothesline before you go to bed.
Here's a youtube video of how this washing method works. You don't need the third bucket if you just place some sticks on the ground to raise the second bucket up off the dirt.
[ame="http://www.youtube.com/watch?v=DKODHAi4Jhg"]http://www.youtube.com/watch?v=DKODHAi4Jhg[/ame]
If you use the above method and you are breaking camp the next morning, you'll still have wet clothes. Just lump them loosely into the cargo area of your rig (inside) on top of your other gear boxes so they are in the open air for the trip to the next camp and they won't turn sour. As soon as you get to your new camp, hang them up to dry as soon as possible. Don't store the wet clothes in a garbage bag or they will turn sour. Don't try to dry them on the roof rack while driving - they will shred in the wind.
If you are camping for an extended period of time with a big group, you'll want to take a few more shirts. Otherwise, it looks to others like you're a destitute bum with no wardrobe

If it's just you or just you and another person, then you don't need more than 3 shirts.
That's it - 3 changes of clothes and you can carry on almost indefinitely.