Would this be an ideal time to put in my two weeks or should I wait
for the paperwork? I haven't been given an official start date yet,
but in the offer details I was given (in writing), the "estimated
start date" would be in two weeks.
You don't have a complete offer yet - you have a contingent offer awaiting real paperwork. Huge difference.
Anything can happen between now and a full offer.
I recently attempted to hire a contractor. In order to start work he was required to go through a background check and a drug check. The first drug test had handling problems (the lab's fault, not the contractor's fault), so it had to be done again. The background check took an extra two weeks due to summer vacations. Long story short - the contractor gave up after 5 weeks of waiting for it to get completed. I don't blame him.
And anything in the company can change between now and the real offer. A reorg could happen, making the position redundant, or having it report to a new manager. Layoffs could happen. Budgets could be realigned. Anything.
Wait until you have a real, written final offer in hand with a real start date.
Then, and only then, give your notice. The wait is almost certain to be very short anyway. The reduction in risk is worth the wait.