I'm wondering why you have ruled out the hybrid option. You can't beat the amount of space you get for the weight. We had 2 hybrids before moving to a travel trailer, they were great. We only moved up to a trailer because of the set-up time for the beds. I love my trailer, but the hybrid was great when our kids were younger and we needed more space in a lightweight trailer.
If you're set on a hard-side trailer, you need to be looking around the 3000lb dry weight range. Why?
As others mentioned, the 5000lb tow rating of the Explorer is BEFORE adding the weight of passengers and cargo IN the tow vehicle. Since you mentioned there will be 6 of you, without knowing the ages and weights of the family and assuming some small children, and an adult spouse, figure 500lbs of passengers before the dog, beer cooler, portable games, pillows snack cooler, etc. A guesstimate of 600lbs of passenger stuff isn't out of the question.
So, your 5000lb tow rating is already down to 4400lbs. Now, take a 3000lb dry-weight trailer, add 40 lbs of propane, 40lbs of battery, 210lbs of fresh water (30 gallons), 250lbs of 'stuff' (dishes, pots & pans, utensils, bedding, clothing, etc.), and another 200 lbs of clothing for the family, and your 3000lb camper now tips the scales at 3740lbs. Now, add another 70lbs for a weight distributing hitch and, the total added weight to your rig is 3810lbs.
Before you know it, your rig and passengers are eating up 4410lbs of your 5000lb towing capacity.
As someone that has personally pulled 98% of what my Explorer is rated to tow (and all my numbers were within spec... barely...) I can honestly say I would not want to do that on a regular basis or on any long trips. I was only hauling my boat 7 miles to the launch ramp and back, but it was white-knuckle all the way and the max speed was 35mph (a function of the route, not a limit of the rig).
Now, in the 3000lb dry range, you're going to be hard-pressed to find a hard-side with bunks. It's probably not impossible, but it's going to be tough to keep the weight down. Heck, even a 12-foot low-wall popup with a slide is going to be in the 3500lb range... A small hybrid might fill the bill as well, but weight-wise, you're going to be up near the limit.
Good luck with your search. If it's not too late, Ford has some smokin' rebates on EcoBoost crew cabs right now. In XLT trim, they'll seat 6 (though they'll be 'cozy', but its towing capacity will easily be into the 7000+ range depending on the option package.
Checkout the Keystone Passport 238ML. I hear good things about it and I'm looking at one myself, I have the Dodge Durango and this fits my vehicle perfectly. It's not a slide, but unless it's raining or I'm sleeping, I don't plan to be in the trailer all time. So while the extra room in a slide is nice, I can't justify the extra pounds a slide comes with when my TV is not a truck.