old oil was white which from what I've read means water has gotten into it
Yes, if the trim cylinders are leaking then water gets in. Since the trim motor was running, I wouldn't take it to the shop just yet, need to find out which cylinder is leaking.
First drain the tank reservoir, clean and refill with oil. Then run the drive up and down while watching the level in the tank. Refill tank as it gets lower. Once filled run the cylinders and find out which one is leaking (maybe both) or where is it could be leaking (hydraulic manifold, or lines)
Is it normal for the outboard to fall like that if there was no oil?
The trim system is hydraulic, and it needs oil or some other liquid to move items. When oil is low, air gets in and air cannot move cylinders