Up front - Axis shocks - very, very good units. You'll have all you'll need to work with, just need to get the spring rates and ride height adjusted. You have the basics needed up front. Those are from the time that Mike Hallock was still making Axis shocks (pre-Penske buy out) and are the generation of Axis that you want on your ride.
Out back - There's a little history about the car's setup and what ATV Racing was trying to accomplish. Sit back and enjoy the read.
When Alain (the guy who came up w/Drakart - from France) first came to ATV Racing (back in 99, I believe), he brought a vehicle that was set up for European style off roading. Smooth(er) terrain, less "suspension use" setups and such were the original idea. Once he got the cars here, he realized that the desert, or out west (where the majority of the car's market was intended on being) was brutal comparatively. He started working with Neil and Jay and talked them into being the US distributor for the Draks. At the time, Alain was in Canada.
ATV Racing, sitting in the northern side of Phoenix, just off I-17, was about 10 minutes from the door of the shop to throwing dirt in the desert. (too bad this area's not open anymore.

- it's now about 20 min. to go play)
ATV Racing had just become a King dealer.
So, taking into account what the car was going to be used for, the new King access and that Jay and Neil are both always looking to improve and make things work better, they yanked off the original shock system and started modding.
Here's a pic from Buttercup (Glamis for those that are unfamiliar with the locals of the Dunes out west), April 2000. This is the very first Drak that ATV worked with:

click on it for a larger pic.
Notice the single rear shock.
So, in an effort to make the car work even better out in the desert, especially through the whoops (if you've never seen desert whoops, you'll have no clue. They're nothing like a motocross track's whoops. Big, rolling and can be absolutely vicious - they're exceptionally punishing) Jay threw on a set of the new King by-pass setups. They were experimenting. Plain and simple.
The dual shock setup, especially on a sub 1000 lb car, (the Drak weighs about 750 - 800, I think) is total overkill. There is absolutely no need for the dual setup.
Yea, it makes the car look like a "real desert racer!" Oohhh, Ahhh

.
They're great shocks (the by-pass) and they look cool and it makes everyone thing "WOW, that thing's bad-ass! Look at the shocks!

!"
Unfortunately they are a total waste of space, an additional 40 to 50 lbs and you'll be a lot happier if you yank them off and focus on the coil over side of the system - getting it set up correctly.
This means getting spring rates right, shock valving correct and most importantly (because you don't want to start tuning somewhere in the middle instead of the beginning of where you should start) getting the shocks set up so that they work with the suspension geometry that you have.
Yes, I know this is not what you want to hear.
"But it has those killer dual shocks on back!". yea, it does and you don't need it.
So, where to start?
The single coil over in back is more than capable of taking care of the rear end of the car. It's a great shock and very tunable. As old as they are, they need to be rebuilt too. It's simple to do.
Let's first start out with your ride height and spring rates:
By raising/lowering the collars on the shock, you'll adjust the preload of the shock spring. This is what sets ride height (provided your spring rate is close). Basically the preload tells the spring how hard to push back in the initial travel. Push the spring down harder, it's going to push the car up farther.
Your car's spring rates should be darn close. Jay and Neil took a lot of time getting it figure out.
So, if you slide the collars on the shocks up a bit (can you get pix of the top of the shocks, front and rear, so we can see what sort of pre-load you have?), you'll be able to lower the car so it works better and handles better on groomed tracks. Yes, motocross style tracks can be rough, but nothing like the desert - drop the car, make it work for the situation - you are not in the desert and the car still thinks it is. You're not loosing suspension movement, just changing where the chassis sits in the movement.
Once you've dropped the ride height and removed the by-pass shocks it will be time to start tuning the shocks. I'm guessing that the coil-overs are pretty much gutted as far as any damping ability. I'd be glad to ask Jay if he recalls what he'd done if you'd like.
Getting the damping set will require a little time on your part - but it will also give you the chance to put a travel limiting spacer inside the shock to keep you from blowing the CVs. The spacer will go under the piston and shim stack and not let the shocks (over)extend.
You'll need to get someone with a camera to help you out. Looks to be no issue, as you've already done a video.
Run the car through the whoops, over jumps and such and get the video person to try to focus on the rear of the car as much as possible - not some close up zoom, just make sure we can see what the suspension and chassis are doing in relation to the ground the car's going over.
Once you get some footage, dialing in won't be an issue. I can help, bdkw1 can help (hint, hint Brian!), Livewire can help and quite a few others can too.
You have the beginnings of a great ride. The older Draks needed some serious TLC to make them work up to snuff, but you're already half way there. It's just time to dial it all in correctly.
Yes, you'll loose a couple inches of travel.
Yes, you'll loose those fancy by-pass shocks (and a HUGE amount of weight comparatively)
But you'll end up with a machine that will do what you're wanting and expecting.
Side note: One thing that Jay figured out was to move the rear lower link mounting position. I can't recall which way he moved it, but axle plunge went from something ridiculous like 4"+ down to about .5". It pissed off Alain that he'd changed the geometry of the suspension ("That is not what I designed!" I heard screamed in the shop as Alain looked over Jay's mods. It was not a pretty scene that afternoon...), yet the mod ended up showing up on the two seaters and later single seaters. CV issues went away for the most part.