What would be the best way to move from Colorado to North Carolina?

I don't have very much to move. 10 boxes at 22x22x22 65lbs max, Honda Civic, and maybe a dresser. I'm currently deciding between Budget Truck Rental, a container thing like u-haul offers, and maybe even mailing the boxes with UPS.
Budget Truck Rental would cost <$1,500. I would have to get a 16 truck to use the dolly and that is a lot more room than I need. Also, gas could cost over $700.
A container would take a while to ship and has a greater possibility for damage. Could cost about $1,491.
Mailing my stuff would cost somewhere around $1100 (is UPS freight an option?). With mailing or using a container, I would drive my car and gas would be somewhere around $200.
I'm in the military and this is a PCS move if that helps.

Added (1). My wife an infant son will be traveling with me.

I do not know about military benefits in this. Several years ago I moved from NC to Las Vegas. I quoted:
Professional mover, POD move, Rental cargo van one-way, FEDEX, UPS, Busfreighter
Your issue in this is getting the car there and the dresser.
since paying for the move myself, I found the cheapest method was getting my car up to full maintenance and driving it, shipping strongly packed boxes on Busfreighter.com with a pickup and delivery. Valuables, breakables, necessary clothing taken in the car. All furniture sold and disposed unless a separate quote for shipping is worthwhile.
https://www.busfreighter.com/our-rates/rates/
https://www.busfreighter.com/
Make sure what you ship is worth the cost to move.
Insurance is minimal coverage.
Boxes are sent in baggage section of buses on Greyhound and get rough handling.
They send a guy with a pickup truck for pickup and then for delivery.
Drive should include gas and maintenance and depreciation wear on the car and generally needs maintenance before leaving, but shipping it is around $700 to $900 on an open car carrier truck.
The dresser is probably $150 to ship alone.
Estimate distance about 1700 miles depending upon city to city.
Two full days plus highly recommended overnight hotel/motel.
$150 gas, $100 additional wear tires etc, $200 car value depreciation, $60 hotel, $50 meals minimum - plan $560 total driving travel cost actual. Most cars examined by a good mechanic will need a few hundred put in prior to 1700 mile drive as people neglect maintenance. Breakdown on the road will cost a lot more. (Not all cash cost, and depends upon needed prior maintenance)
Busfreighter estimate
10 boxes. 550 pounds total, $1000 insured value
80909 to 28210 sample
7 days door to door $542.44
Your total move estimates at about $1100, dirt cheap by current standards, plus cost to move dresser.

You are probably underestimating.
You don't have enough stuff to fill a pod.
Shipping by USP/FEDEX at $1100 makes sense, and Busfreighter at half that.
Car cost is underestimated, and overnight driving is hazardous in the winter.
You also left out moving boxes and packing material supplies will run about $60 (Home Depot heavy duty large and extra large, though some can be standard not heavy duty).

Have you looked into getting a smaller truck - one more suited to your need - and having your car transported there? I have no idea what car transporters charge, but that combination might be less than a larger truck and a car dolly.