I usually clean the inside of cables with WD-40 first, and then flush that out with a few sprays of brake parts cleaner. Then swing the sumbitch over your head rodeo style to clear as much of that as you can out.

One more thought on the sticking slide. If your carb is screwed onto the manifold too tight (beyond where your elbow pops once), the you can actually distort the shape of the barrel of the carb, and even a couple of thousandths can cause the slide to bind in the barrel.

Make sure the throttle grip is not hitting or rubbing the handlebar end. Loosen one of the phillip head screws on the throttle housing and slid it out a bit. See if that doesn't help. While you are at it you can spray some silicone spray on the throttle tube that rides on the handlebar. Clean it first with a can of brakleen if dirty or gum up. You want to use a non petroleum spray or lubricant. Silicone spray is your best friend on these bikes. 

By the way, Super nice looking CT3! 

 
Mothersbaugh that imagery gave me a good chuckle at work, thanks!

I took the housing off of the handlebar and cleaned out the throttle tube and that metal tube part that's close to it. Big improvement! Thanks all for the advice!

And thank you SkyDancer, she's "cherry" as my brother said
If it wasn't raining tonight I would have taken her for a quick little ride, so instead I just fired her up in the garage and sat on her for a bit after getting the throttle issue fixed. Getting plates and insurance on Friday so I can go riding with my friend on her 250 ninja
 

Just putting science to work for me....

"What do you suggest for cleaning? I currently have carb cleaner and WD-40 in the garage. Also, clean where exactly?... Sorry if that's a stupid question, bikes are still pretty new to me."

Undo the screwtop of the carb, remove slide (aka valve) and give it a gentle clean with Autosol. Do the same with the inside of the carb, where the slide "slides" up and down. This is best done with the carb removed, so as not to leave gobs of Autosol in the carb.

Saying all of that, it now looks like you're sorted!

If you haven’t changed the brake shoes yet please do so. They can delaminate and lock up the wheel at the worst possible time.
Very nice bike.

My name is Dale and I'm a mechanical engineer from Vancouver BC, Canada. I just picked up a mostly complete 1972 DT2 from a buddy of mine that had too many projects in his garage and was willing to give this one up.

My current regular ride is actually an '89 Honda CB-1. I'm looking forward to the simplicity of the 2 stroke single for a change. Previous bikes include '82 CB125S, '81 RD350, '73 CB500 Four and wayyyy back in the day my first bike as 14 year old was a Rockford Tora 60 cc that died so I substituted the engine from an XL 75 from a high school friend. That was early Eighties.

Looks like this bike is mostly complete with a lot of parts in bins. The engine needs a piston and cylinder re-bore but it turns over fine. Electrics are all there but need re-doing. My present plan is to just to a basic restore and convert to 12v. I will tear the engine down to check everything before rebuilding. Tank is rusty and needs to be sandblasted, sealed and re-painted. Forks are a bit pitted too.  It will need lots of other little things but I have time. First thing is to get a service manual.

I'm all open to suggestions for restoration steps and recommended order of rebuild.

 

Welcome from NZ. Bike looks like a good one to start with & a few spares as well. Not a spare 360 motor, the other bits n pieces?.

Nice! 

from the UK.

 
The Tech Library has a service manual for your bike. Good luck with your project.