. How do I find out whose fault it is?
I wouldn't be concerned with that. If it was a common element, then the HOA is liable. If they believe you are liable, let them show why.
1. the main reason the insurance company drug out the payout for so long was because they felt the flooding would not have happened if the HOA had posted information reminding tenants to turn on their faucets to drip
that is exactly why they shouldn't be coming after you. If it was the HOA's error in not requesting you allow some water to flow, that was the HOA's error, not yours. Tell them it was the HOA's error for not asking you to run some water so go after the HOA for the deductible.
Besides that, I do not believe even if they had given a directive to allow water to flow it would have relieved the HOA of liability. It is the fact the pipes are not properly protected from the weather that caused them to burst. Not that you had not opened your faucet. What would they do if you were on vacation in Europe over that period of time?
So, the construction of the common elements is not within your control. While it would have been great if, given this is a very unusual situation, you could have prevented the damage by opening a faucet, I do not believe you are liable for anything because you didn't.