Why do you need to drain?

High nitrates or phosphates can surely feed algae, but with proper chlorination, algae wont form.

There is a super handy chart at trouble free pool dot com.
For some reason this site is blocking me from posting the link, which is nuts, that pool site is incredible and I have never seen spam from them.

try googling "cya chlorine chart"

The chlorine binds with the nitrates and becomes inert - we have shocked the pool and the chlorine is still at 0. We can keep it this way but would use a lot more chemicals to maintain it.

If we did not hit water when we built, are we probably going to be ok with the water table now?

It even says on that website
"At very high CYA levels, over 80, a partial drain/refill to lower the CYA level is usually required."

You didn't finish shocking. Something is in the water. Anytime free chlorine combines with something, be it
bio-badies or tanning lotions, it becomes inert.

Nitrates mean zilch. Killing the algae before it can eat solves any issue there.

If its a bio-badie, the time spent draining and refilling will give the bio-badies the opportunity to reproduce.

If you have a salt cell, your CYA is perfect at 80. If you are tablet fed and not in the Southern latitudes, then diluting will be needed to reduce the CYA level.

With your current level of CYA, bring the FC level to about 40 and hold it there until you can pass the Overnight Chlorine Loss Test. If you can take the calcium, cal-hypo can be used. I prefer using bleach (6%) or chlorinating liquid (12%).

Scott