I think kmbelt nailed it.
You can find travel trailers (used) for a $4000 budget. Most will probably require some work, and all--even brand new ones--require maintenance. Finding one that will somewhat comfortably sleep four and be suitable to be towed with your vehicle and be in decent condition will be rather difficult, I suspect.
A popup would, in this case, I think give you more space and easier towing and be more budget friendly. They do, of course, have disadvantages in terms of amenities and comfort, but are a definite step up the luxury scale from a tent. A somewhat similar option might be some of the collapsible hard-sided trailers, like the A-Liner, though most of them would be a little smaller.
Another option to look into might be renting a motorhome.
North Carolina looks to have pretty reasonable taxes and fees. The highway use tax is 3% (due at title transfer), or very roughly $100 for your price range, and other fees might bring the total up to $200ish...if I'm reading their web site properly. Calling the DOT information line would be the best way to get accurate information.