Hmm...!
Well, whenever I want to memorize a piece, I do it in short segments. I take the score, then I play a short segment of the RH and then I immediately close my eyes and try to play it mentally. I literally just imagine my hand playing it on the keyboard, with the music. Then I repeat the same with the LH, playing a short segment and then playing it in my head. If I suddenly get confused when trying to play it in my head, I just play the same segment again on the piano. This time I will do it slowly, while paying careful attention on how my hand and fingers move and what keys they press. It usually works and does not take long at all.
I will just repeat this procedure until I have the entire piece memorized, then I stop practicing. It might take an hour for me to memorize a new piece, depending on the length of it. So, after I have quit practice, I will try playing it mentally to check how well I have memorized it. For long pieces, I almost ALWAYS forget one little thing!
It does not matter as I will just rememorize the forgotten part the next time I sit on the piano. I usually never forget it after that.
Um.. maybe I will sum up the method?
1. I memorize a short segment, usually consisting of 2-3 bars or something like that.
2. I pay careful attention on how my hand plays it and how the music sounds like.
3. I play the segment mentally, imaging my hand playing the same segment with the same motions as I would actually play on the piano.
4. If I get a black out when I try to play it in my head, I just repeat it by playing it again this time on the piano, paying more attention this time. It usually takes less than 15 seconds for me to do.
5. I repeat the same process, this time for the LH.
6. I repeat this entire procedure until I am finished.
The advantage in this kind of memory is that I do not remember the music as tadpoles on paper but instead as black and white keys on the keyboard! All the expressions such as crescendo become mental concepts of the music which I simply hear when I play the music in my head. I just memorize the music the way it is played on the piano, not written on the sheet of paper!
You said that you have a problem with the LH music often making no musical common sense when played alone. I have no problems with that when I apply the above method. Maybe it will help to just accept the music the way it is?

I do not think you need perfect pitch in order to memorize. I don't have it myself and I am able to memorize easily.
I hope my advice helped you in anyway! I tried my best!