Here's another vote for actually writing out the arrangement. On paper. Something about taking a pen to paper really ingrains it in my head.
My arrangements look like something like this: Intro/V/V/Ch/Lead/Br/Ch/Ending. I'll notate what beat to play, what time sig, tempo etc.
I don't bother writing the number of bars in each part, because I have a good memory for that stuff once the song is rolling. Unless there are really tricky parts, then I do notate that.
But it's vital that I have a clear map of arrangement the song. Once I actually write the arrangement down on paper, it's for the most part memorized.
Personal story...I did a fill in for an original band a few years back, no rehearsal, probably 20 songs. Did I mention that I had no rehearsal and all original music I had never heard before? I did have access their music online so I was able to map out all the arrangements. I just came to the gig with my stack of my arrangements written out on big index cards in black sharpie.
The agreement was to play their songs in a certain order, and I stacked my index cards accordingly. Nailed the gig. Nailed it! Couldn't have done it without my arrangements. One really useful trick I learned when mapping out an arrangement of a new song that I had never heard before...is to write on my little card that this song has the same groove as (insert similar song here) That little reminder helped me greatly to remember the song before starting it.
TL;DR: Map out the arrangement
I do the same thing, and go a bit further by writing down the first line of every verse after the letters. So my cheat sheet for Sweet Home Alabama" looks like this:
Intro - no drums; guitar riff 2x thru
V1 - Big Wheels...
V2- Well I heard
C
V3- In Birmingham
C
V4 - Now Muscle Shoals
C
Solo-Bill (guit)
Solo-Bill(sax)
C - 2x and out
in my original bands, and my cover bands, I totally organize where I am in the song by knowing the lyrics. I also can sing the first chunk of most of the solos as well, which helps me know when they are coming up. That way the song is "going through my head" as I am playing it
on the other side of things...the more technical side...it sounds like you need to NOT do full run-throughs of the songs in the wood shed at first. If execution of certain parts is the issue, single out and rep those parts first...then try the whole song. This process will also help you develop some "recovery riffs" to use when you do get off track
I remember when trying to learn many of The Police songs back in the day, I had to single out fills and learn them first, before trying to get through the whole song...but i could pretty much play a Motley Crue song all the way through for the most part, and if I missed one of Tommy Lee's fills note for note, I had one 'in my toolbox" that I could pull out quickly in a run through
I play in 2 band situations right now where there is very little practice, and a lot of "audibles" called on stage, so using the list system above, and the woodshedding process mentioned helps me survive those gigs. In fact, the show Saturday I had never even met the guys I played with, and when I asked about a set list, the answer given was: "if you have listened to the classic rock station in town, you will do fine..." man, my OCD alarm was going off all week !!! But in the end, it was fine...in fact, it was pretty freaking good!! I just had to learn to trust my self!!!