If you had a sinkhole in one spot, it is possible there are more you cannot see. I would bet the problems did result from dry construction leading to poor compaction.

I am no expert, but I think it should be fixable if you are willing to drain it. You need to find the leaks and or re work the lining with proper moisture. Maybe bring in more clay as needed.
Pay once for a good fix, or pay to pump forever.

As for the builder, that doesn't sound good. Depending on any agreements on paper, it may be time to call a lawyer if the fix expense is high.