UPDATE:

It turns out that I had overfilled the unit. I filled it with oil in
the fully "up" position. When I dropped the outdrive to "level" and
opened the oil vent plug, several ounces of oil came out. After I let
out all of the excess oil, no more oil coming out of the drain plug.
While I take this as a good sign, I still suspect that I have a bad
seal internally. I ran the boat in the water yesterday and I'll check
the level before putting it back in the water. What makes me think I
still have a bad seal is that one of the diagnostic tests for a bad
seal is to pressure test the unit. If a good unit holds pressure then
that means that any connection of the oil reservoir to that drain hole
is not normal. I think that I've stopped leaking oil because it must
be one of the higher seals that is bad and now my oil level is below
the seal.

Does this make sense?

I also have milky oil every time I check the level. I feel like annual
replacement of the oil and checking the level every time I put it in
the water will let me run safely indefinitely but how much should
professional replacement of all the seals cost? How about if I pull
the upper and lower unit off myself and bring it in?

Chris