It is most definitely large enough for a week of camping for two persons.  You'll have to leave the kitchen sink at home.  And as a ppersonal preference, on such excursions learn to live without a cooler.  If you're bringing food for a full week for two, or especially if you intend to carry all water from start to finish, then you'll have a full boat, but plenty of room if you pack light.  Heck, backpackers do it all the time with a 40 lb pack (maybe without the full week's worth of water), so think along those lines, and you'll still have the luxury of doubling that load per person.  I've done it may times in a 17 ft Grumman canoe (2 + dog), just never in my dory.  Carrying capacity probably about the same.

You'll find all sorts options for rigging up to sleep aboard / tent over the boat here on the forum, if you desire that - and if the two occupants are friendly, sleeping aboard for two could work.  I would sincerely hesitate with anything like "living" aboard - just not much elbow room for cooking & etc. with the rest of your household all on board.  Things like that probably best done ashore.  You'll need to stretch your legs - althought relatively stable, you won't spend much time standing up in a NE Dory.  I only mention sleeping aboard because dependent upon plans and location sleeping aboard could be a drier and less buggy option, and less chance of fights with racoons or gators or bears.

Everything best in small gear-boxes and dry-bags.  Don't throw stuff in loose, especially if you'll be sailing.  If you intend long days of rowing, figure out a system: a mirror, your partner giving directions, a stern-mounted compass (that's my best solution) and practice sighting on things astern, allowing you to minimize the number of times you need to look over your shoulder.  That can become annoying at the end of a long day.

   Oh, and let us know how things turn out.  But if your question is based on the fact you are just starting to think about ordering a kit, maybe your trip report won't be forthcoming any too soon.  But (with the ability to work on the boat a full week during Thanksgiving and 2 weeks during Christmas time-off breaks - yes I work in education) I finished my dory in less than 3 months while having a full time job, including sail rig and some custom seats/floatation mods.  But everyone works at the pace that fits their lifestyle (and obsessive-compulsive tendencies).

Yeah, it should work, at least if your able to camp ashore at night.  Like with learning to sail, you just have to work your way into it by degrees.  Start out with short duration trips, maybe staying in the same place for a couple of nights and then daysailing from your camp until it's time to go home--sort of "cruising in place".  You'll figure out what works and what doesn't as you increase the duration and distance you mean to cover, maybe working your way out from and back to your take out point in "petal routes" so you'd have opportunities to cut the trip short if the weather turned foul or some such.

Some tips, in no particular order:

• Keep the cooking simple.  I got to the point where, even aboard a 19', 2900# catboat, I didn't plan on doing a whole lot more cooking that what could be done by boiling water.  Put the kettle on for hot beverages, instant soup, or noodle cups, and you can make a hot meal out of packets of variously seasoned tuna wrapped up in tortillas to go with that.  You can heat up cans of stew, spaghetti,  heartier soups, etc., by setting the cans in hot water for a bit, and thereby spare yourself some mess.
• I'll second the motion on the no cooler idea.  In my opinion, more trouble than it's worth.
• Keep the clothing simple, too.  My then eighteen year old son and I cruised for over a week in the North Channel (Lake Huron) in a Sea Pearl 21 (a long, skinny, mostly open, cat ketch with no hard cabin) with two changes of quick-drying clothing each.  We'd change into the clean set and wash what we'd taken off, spreading them out to dry as we were able during lunch and supper breaks.
• As noted above, think like a backpacker, only you can carry canned food and a lot more potable water (recycled half gallon juice bottles work well for that) than you ever could backpacking.

Gosh, it sounds like fun!

.....Michael

   Thank you for the comments. Most of my gear I already have is for backpacking, mountaineering and kayaking so keeping the gear lighter and compact isn't an issue, I've just never seen a NED to be able to judge the whole sizing thing. Just making sure before I start it or need to switch to a bigger choice. Your help is greatly appreciated.