You're stuck because there really isn't a way around lowing that CYA level without dumping a lot of water and diluting whats left. There are two main options. One is using a really big tarp over the pool and adding fresh water into the tarp into the pool as you're pumping out the old stuff from under. The advantage to this is it's pretty much a one for one exchange and if water is expensive or scarce it can pay for itself.
The other way is to add fresh cool water into the bottom of the deep end and pump water out with a portable pump, sump pump ... from the shallow end at the same time. The weight of the cooler water will to a point help separate it from the water you're pumping out but it still mixes some.
If there was another way we would tell you as we all know the pain in the butt it is to throw away that much water. With a high CYA level it's what you have to do if you want to get any control of your pool. Even a pool company is going to tell you the same thing or if they don't they shouldn't be touching your pool as they can't really control your chemistry with the water as is.
Chlorine tablets and granular chlorine will contain some CYA almost always. Please don't add any more of that stuff to your pool or you'll start going above 200ppm CYA shortly. You want to start to use only liquid bleach or liquid pool shock (bleach) as a chlorine source as it will not raise your CYA. It's likely also cheaper so that's the good side.
Once the CYA is down to 70 or less would be even better you need to SLAM the pool. It's like shocking except you only need to do it once and then just maintain your pools chemistry instead of weekly blindly tossing in bags of shock to try and keep the water from turning green.
You need to keep reading the Pool School articles and the recommended chemicals. You'll see our way is not the pool store way and it's likely not the way the pool guys you're trying to hire would do it. The SLAM takes some time and patience but after it's over our way is easy and much cheaper that running to the pool store and buying an armful of chemicals. In a few weeks you'll actually understand what does what and why you are adding it and what to avoid like algaecide.
You will need to buy a good test kit and that's the beginning of tuning the water sparkling blue and having a sanitary pool to swim in and enjoy.
TF-100
http://tftestkits.net/splash-page.html
K-2006
http://www.amazon.com/Taylor-Complete-FAS-DPD-Water-K-2006/dp/B0002IXIIG