There is really no way to tell how much you will need to drain. Start with 3ft of water but be ready to drain more if you have to....More than likely its the shock or tabs you are using that are increasing your stabilizer levels....Avoid quick shocks they are full of stabilizer...

There are extended test kit directions as a sticky on Trouble Free Pools site that will show you how to dilute your pool water by half before testing your CYA (then you multiply the test result by two to get back to your "real" CYA) if you have your own test kit - this might work with the pool store's testing (I'd do my dilution before I take it in and not mention it to the store - they often like to argue with me about TFP and BBB and the rest). I ordered my own test kit and got it within a few days and am so happy I don't have to rely on the pool store's guesses on my CYA now, so would highly recommend that.

Your target CYA is different if you have a salt or regular chlorine pool, but you can plug numbers into the website the pool calculator dot com and see how much it recommends you drain. However, it's only as accurate as to the numbers you can provide, so getting a solid CYA first would be helpful. I plugged your 23000 gal pool in, and asked it to take a CYA of 180 down to 50 (where I try keep my non-salt, full-blazing-sun pool with BBB method) and it says you need to drain and refill 72% of your water. I would not do that all at once, I would drain some and refill, drain and refill, in several cycles. Pools have a way of popping out of the ground if you don't keep water in 'em.

There are several threads on TFP related to water replacement due to CYA, and lots of helpful folks to give advice.

Also of note - to a certain extent, my understanding is not necessarily that your chlorine is ineffective at high CYAs, but it's more difficult to keep the chlorine high enough to be effective at the high CYA levels, but I am science challenged so I don't know for sure if that's exactly right. I do know, the higher the CYA, the higher your Free Chlorine must be. There is a chart showing the relationship on TFP as well, it goes to CYA 100 and says FC must be at 39ppm to be shocking the pool. If your CYA is 200, your FC might need to be around 80 to properly shock the pool, which might be nearly impossible. Pool School will tell you how to properly go through the shock process once you have the CYA under control. Good luck!

Thank you for the information. We drained about 40% of the pool, did the exact dilutions and took it in for another reading and it was between 180 and 190 using a more specific CYA test. So we are continuing to drain....darn! But the advice was great, so thanks for both of your answers.

There is a website www.poolcalculator.com that will give measurements for all of the chems that a pool might need. You can do a CYA measurement on the site as well and it will tell you how much has to be drained. With 23K gallons and a CYA/PPM of 186, you would need to drain 78% to get to 40ppm which is the desired CYA level.

When you are done with the draining and refill, you should move away from pucks for sanitation or you will put yourself right back into the same situation in a few months.

Are you sure about the CYA (cyanuric acid) reducing the effectiveness of the chlorine? I've never experienced or heard that. CYA is a stabilization additive that reduces the rate of chlorine evaporation due to the sun.
In a 125,000 gallon pool, we use granular CYA and liquid chlorine. Keeping the CYA level between 120 and 150 reduces our chlorine usage to about 1/3 of what it is at a CYA of 30.
We do, though, shock and filter the water until it's crystal clear before adding CYA at the beginning of the season.