I don’t think that Morrison’s is a particularly easy tune to remember (nor a particularly easy tune generally - it’s just somehow gotten into the canon of beginner tunes in the US).
As for memorizing tunes, I think the point is to break it down, to understand the structure. Mind you what I am telling you now is reflects my level of understanding, which is limited. There are two basic levels to a typical tune: a question/answer structure, and the individual phrases. Take a simple tune like the Burnt Old Man:

(That’s not quite how I play it, but just what I could grab off thesession.org)
The first question covers roughly the first two measures (note that bar lines and beginnings of phrases fall in different places in this music):
[FE] | DFB AFD | ~F3 F
(you should think of this question of containing a lead in, even though it’s not written). Then there is the answer:
FE | DFB AFD | ~E3 E. Think of this as Answer 1.
The question is repeated (but goes up to the high d - many jigs repeat the question 1:1), and then is followed by Answer 2, which also wraps up the part:
de |fed B2d | ABA F .
Look at the B part in the same way. Again there is a question-answer thing going on, with Answer 2 being the same of or similar to Answer 2 of the A part.
That gives you the basic structure.
Break down each question and answer into the phrases. In a jig, a phrase straddles bar lines, like so:
FE | DFB A;
next phrase: FD | ~F3 F; and so on…
Listen and practice tunes by phrase, until you have the question. Then learn the phrases that make up the answer 1, and so on. Soon you’ll be thinking in phrases and will understand how players change phrases around to make them fit together (try playing the A part of the Burnt Old Man repeating the question exactly and then going into the answer 2 - it works: … EF | DFB AFD | ~F3 F de | fed B2d | ABA F).
Once you hear and play phrases, you put together the questions and answers and then thing flows and makes sense. When you learn tunes by ear, you listen for the phrases and learn those. Find the “heavy notes” in the phrase and how you get into them and out of them.
Don’t use sheet music until you can hear understand phrases, questions/answers, and heavy notes (my term. That is, important notes - the bones of the tune). Sorry to break it to you, but I don’t think your “going over the plot first with sheet music” thing works.
FWIW.