The best way I've found to memorise tunes is practice, practice, practice... obviously. But, more specifically, one useful way I've discovered for memorising tunes is to distil them down into their most basic harmonic structure. Take a simple Jazz Blues in F as an example:
F7 |Bb7 |F7 |C-7 F7|
Bb7 |Bo7 |F7 |A-7 D7|
G-7 |C7 |F7 D7|G-7 C7||
Now, break that down to Roman numerals (if you're not sure about this, stop reading and go find out about Roman numerals and chords).
I7 |IV7 |I7 |V7 I7 |
IV7 |#IVo7 |I7 |iii7 VI7 |
ii7 |V7 |I7 VI7|ii7 V7 ||
Or:
1 4 1 5-1
4 #4 1 3-6
2 5 1-6 2-5
Now, this not seem any easier to remember, but it's the little things that count. Instead of trying to remember what the actual chords in the last two bars are, you just remember it ends in a two-bar 1-6-2-5 turnaround. Kind of like a phone number. And the most useful part is that if you know the basic harmonic structure, you can transpose it into any key.
This is how I managed to memorise Rhythm Changes all those years ago. I always seemed to have difficulty getting starting; once I was going it all came back. So, using this method, I just kept it in the back of my head that Rhythm Changes (generally) starts with 1-6 2-5 3-6 2-5. The whole middle section is just a bigger 3 6 2 5.
Now, with regard to walking, I would suggest going back to basics.
I used to do something I called 'sequencing', inspired by the way a lot of horn players seemed to have a catalogue of licks and melodic lines that would reoccur in their solos in different songs - think Charlie Parker.
Pick a chordal or 'scalular' walking pattern (like 1-3-5-3 or 1-5-1-1/2 into the next 1) and walk over an entire head with only that pattern. Then pick a new pattern and do it again. Get more complicated. Then do alternate bars. Then do cycles of four different patterns over four bars. Then add rhythmic embellishments. And so on. However you experiment with it, the basic building block should be predetermined one- or -two bar walking patterns.
The idea is that you build up a store of very short but very solid walking lines that you can draw on, combine, and alter on the go. At the very least, it means you can have a whole lot of 'safe' lines for when your feeling unsure about what to do over changes.
My 0.00002c.
EDIT:
This site has a good list of some pretty ubiquitous jazz standards, plus the changes for them in a couple of keys -
http://www.learnjazzstandards.com/index-of-jazz-standards/index-of-jazz-standards/
Some that I would get started on straight away:
Jazz Blues in at least F, Bb, and G (covers a lot, like Billie's Bounce and Now's The Time for the most obvious examples)
Rhythm Changes (which is the basis for a truckload of other tunes - Anthropology, Cotton Tail, Dexterity, Oleo, and on and on and on)
Autumn Leaves
All The Things You Are
Blue Bossa