Factory tow package would have come with a class III/IV 2" receiver. The fact that you had to put a class 3 hitch on your Ex implies you do not have the factory tow package. This means you only have 3500 lbs max tow rating, and 8,000 lbs GCVWR.
4440 (curb weight per Edmunds)
3146 (loaded popup)
400 (family weight ???)
7986 lbs (Total/actual GCVW)
As you can see, you are right at your maximum. Your rig is fully utilized.
In order to have more towing capacity, you would need to retrofit the components that came wiht the tow package:
Class 3 hitch (already done)
Aux transmission cooler (I recommend you add this even with the popup)
Rear axle ratio change (from 3.55 to 3.73)
Axle change is probably cost prohibitive compared to trading your truck for another that already has the tow package.
A V6 with tow package will again be fully utilized (no cushion) with a trailer that size, as you will be 4800 to 5000 loaded. But the V8 model starting in '06 gets better gas mileage than the V6 with tow package. This is because the V8 gets a 6-speed automatic trans that does not requires the gas-hungry rear gear change to get it's full 7000 lbs rating. All the V8 tow package includes is the bigger hitch, extra wiring for brakes and 12Volt, and the trans cooler.
So, either:
1. Keep what you have (and consider a trans cooler)
2. Get more tow vehicle.
