LG, I know it is months later, (just saw your post) but yesterday I just did basically the same thing you did. (same outboard you have) I had an issue on my last fishing trip on the Mississippi, where the outboard didn't want to tilt up. It did, but slowly. So when I got home I started working on it. I have the owners manual for the Outboard, that came with the boat, but it is pathetic to say the least. I, like you decided to drain and replace the hydraulic fluid completely as it was apparent that this boat had not been maintained properly. When I pulled the bottom plug, which the owners manual said to do to replace the fluid, I also got blasted, when looking in the hole I saw an "o" ringed hole inside a ring that had 4 slots on the outside edge.
My machinist's mind said hum-mm, why the O ring? So I started looking immediately on the ground and found a small beige nylon needle valve (looked like a large float bowl needle) and a spring. I cleaned these and set them aside on the top of the casting deck. I let it finish draining for an hour or two. I then put in the needle valve, then the spring, and then the bottom cap. Before I had started I had raised the tilt all the way up and set the tilt lock to hold it up. I read this thread and no one mentioned these parts, and with your having gotten a fluid bath like me I figured you also shot your valve and spring out of the hole.
I filled the reservoir with new oil until it began running out the top hole, then closed it up. Hit the up and down trim buttons but it did nothing. After much fiddling around, and having noticed the small screw to the left of the bottom oil drain cap with the retainer snap ring, I figured it must be some type of valve, I opened it a small amount and hit the down trim button, I could hear a gurgling sound as the oil was being pumped, but knew that it was air I was hearing. What I finally worked out to both eliminate all the air in the system and get it full and working properly was this following sequence.
1 Close the relief valve and fill the reservoir. (bottom left screw with retaining ring)
2 Activate the up or down tilt buttons and then open the relief valve while running the tilt, until you no longer hear the gurgling, then close the relief valve.
3 Repeat until your tilt begins to work. Alternate between the up and down tilt buttons to fill the hydraulic system and get the air out.
4 It may take awhile, and more fluid than you think. I probably (didn't count) did this at least 15 times to get the air out and completely fill the system with the new oil.
5 As someone else mentioned let the system "rest" for 10 minutes or so after you fill it several times to let the air bubbles rise out of the system. Each time I opened the top plug to add more oil, it hissed air and the oil was white with foam, so take your time and allow the tiny air bubbles to come to the top of the reservoir.
Each time I ran the tilt after adding oil I ran the tilt for probably at least around 10 seconds. It took awhile but my tilt is working perfectly and holding position again.
HTH