Talk to the leader about getting the songs for the next two weeks every rehearsal. This will allow you to work on them a little longer and less intensively so that you can work in practice time for other things.
I am in a similiar boat in that we 6 to 9 songs a week. We only rehearse 2! I have to play the rest from memory or on the fly as we go. No charts (i'm the only musician who can read any form of music). Not sung to the record and some songs are simply traditional and vary from singer to singer. (ever tried to play behind people singing the same song in different keys? LOADS OF FUN
).
Hang in there it gets easier as you go along. I've been doing it for 5 years or so and you learn to adapt. After a year you will probably find that the songs follow similar patterns and the new ones will be easier to pickup and your practice time will return. As someone said the best practice you can get is playing with others so the year or 2 that it takes to get this under your belt will leave you with a great foundation for the next leap you make.