Hi Welcome to the club @jimthfc. I mean this quite literally, in the sense of welcome here in the community. But also in the club of those that are challenged by songs and the wish to memorise them by heart.
First off, it is a good thing to memorise songs, or so I think, because it allows you to keep the rhythm going. Usually, one look at the chord progression will help you get through an entire song, because they repeat. If you have to look each and every time at the leadsheet (where am I) you are bound to lose the groove and build in mistakes, and you lose track of where the others are when you play together.
Obviously, when I read the title to your post, I thought you meant memorizing lyrics, but the other dimension is memorising chord progressions and rhythms etc. On the former, I am challenged, and I am happy to limit myself to a maximum of four verses, and a single chorus, which is the limit of my memories capabilities. Not for me a Folk Ballad like Matty Groves with 12+ verses.
On the latter, you’ll have to develop muscle memory guide you. Obviously, start with songs that have an easy groove that you are very familiar with and with a very easy structure, like a blues progression or something similar, involving 2, or 3 or 4 chords at the most. Play it often. Then hide the chordsheet, and try and pull it back from memory! Then, try and do that in the dark! It helps of course if you have something of a backing tape to play against, certainly.
When you are safe on the basic rhytm and chord progression, you can build in the intro’s/solo’s outro’s/embellishments. Leave complicated songs with different verses , choruses and bridges and solo parts for later, you are bound to set yourself up for al lot of frustration.
Even if you don’t sing, it is nice to know the words in the song, because then you would not lose track of where you are in a song, and when to transition into a bridge, or prepare for the outro.