You've received excellent and detailed advice in response to your questions. Most of we RTA regulars have "been there, done that", and often for decades. Build some 2 day stops into your itinerary. You'll be glad you did.
Aside from that, if you still need to cull some drive-bys from your plan, consider how hot it may be in the lower elevation portions of Utah and West Texas, especially Big Bend. Even in late May-early June, camping may be very uncomfortable and hiking downright dangerous if the weather is particularly hot.
After some more planning my girlfriend and I have made a very rough and flexible schedule
Columbia to Nashville
Nashville to Kansas City. stopping in St Louis along the way
Kansas City to Denver
Denver and Rocky Mountain NP
Denver to Arches. Camp and hike.
Hike Arches. drive to Bryce Canyon and stay the night in a nearby city
Bryce Canyon to Zion. set up camp and hike
Zion to Horseshoe bend to Mesa Verde
Mesa Verde to El Paso
El Paso to Austin
Austin to Alexandria LA
Alexandria to Columbia Sc
That's better but there are still a couple of problem spots.
You really have no time to see any of Mesa Verde, despite your plan to stop there. Day 9 will mostly be in the car - especially factoring the stop at Horseshoe Bend, and to see the most interesting parts of Mesa Verde, you need to take a guided tour, which will almost certainly be sold out for the day by the time you get there. Meanwhile, day 10 you'll need to be on the road early road if you want to make it to El Paso the same day.
The stop in Alexandria LA is puzzling. It's not on the direct path, and it's way too far away from Columbia to make it as your final stopping point. If you try to drive from that 800 mile section, especially on your last day, when you'll certainly already have some level of fatigue, you're just asking to get into a serious crash. Austin to Columbia is the very top end of what you can do in 2 days, and for that you need to stick to the direct route and stay right at the halfway mark - which is around Biloxi, MS.
I also don't see the point in visiting RMNP as a day trip from Denver. It would be at least 5 hours of driving without moving forwards that day. Why not look at staying on the other side of the park, Granby for example if the Trail ridge road is open or around Idaho Springs/Georgetown if it isn't. Either option would at least give you a little more time in both the parks and save unnecessary back tracking.
It takes an hour or there abouts just to get to the visitor centre, once you have entered Mesa Verde. It needs at least a full day, if you are able to book tickets for both tours on the same day, to appreciate the wealth of that park. If you can't spare that amount of time, I'd leave it for another trip, when you have more time.
A couple of comments on Mesa Verde NP:
1) From the west you can purchase tour tickets at the Visitor's Center in Cortez, CO, about 10 minutes from the entrance to Mesa Verde. Lines are usually shorter than at the visitor's center and they are directly hooked into the same booking system.
2) The Visitor's Center is now very close the highway/entrance to Mesa Verde NP (it used to be a half-hour inside the park)
3) The campground is about a ten minute drive into the park and is very large. You can pay for your campsite at the camp store. Showers are included in the nightly camping fee. Pay at the camp store and then go pick your site.
4) Great brew pub in Cortez, CO, Main Street Brewing. Good authentic German brews on-site and American style brews. Very decent food as well.
Recommend spending two nights at Mesa Verde, the first being just an arrival night and the second night after spending a day in the park.
Another thing to consider, right now, the last 5 days of your trip are essentially all driving days. You've mentioned places you want to stop, but realistically, you're going to be on the road almost all day every single one of those days. Not only do you not really have time for Mesa Verde, but the stop in El Paso you are going out of your way for is another case where you'll arrive late and have to leave early the next morning.
I mentioned it before, but it bares repeating. Going down to El Paso and Austin after Colorado adds a solid 500 miles compared to just heading home on I-40 - so you really are adding one full extra day of driving. Considering that you don't really have time to stop for anything on that section, it doesn't seem worth it to me.
A couple of comments on Zion NP, Bryce Canyon NP, Grand Canyon, and Mesa Verde.
Elevation is a temperature factor (and a breathing factor as well). Zion is at a lower elevation than Bryce meaning it becomes a lot hotter during the warm weather months. It is worth while to check elevations!
The Grand Canyon North Rim will be open and is along the way way between Zion / Bryce and Mesa Verde, especially your planned routing. The North Rim is great! Bryce to Mesa Verde is a stretch in one day. North Rim campsites need reservations far in advance, as soon as they go on the market.
If you can expand from 14 to 20 days this area is where I would make the allocations.
thanks for the responses. Mesa Verde is the park I have done the least amount of reading on so far but from what you guys are saying we will probably stay the night in or near Albuquerque instead and then drive to El Paso after that. We have family in Alexandria so that's why the stay there. And I have done the drive from Louisiana to Columbia a few times so im not that worried about that.