Update:
When I told the park's manager of the problems I had, he fully admitted that his people had provided terrible customer service, he made no excuses, and he said he would speak to the person responsible as well as his entire staff. He also offered to me the first of my 3 nights for free if I would give them another chance. I agreed to do this.
So does the manager's actions make it "all better"? While I give him full credit for taking responsibility in a professional manner, that doesn't clear the slate. I should have been able to make one phone call, reserve any available site, and receive an e-mail confirmation. It remains that I had to make three phone calls and still was unable to get a confirmed reservation that I needed. (After a fourth call (this one to the manager), I believe I now have the reservation I wanted.)