Congrats on the position and the gig. I can't speak to the best way to approach this, but I'll talk a little about what I do.
I've always been an ear player, and so I can pick stuff out without charts and such. However, I find that I'm not always good about remembering changes until I play the song with the group. So sometimes, what I'll do is import a track into my DAW of choice to count out the measures if each part. This lets me know that a verse is 16 bars, a chorus is 8, there's a 32-bar bridge, etc. It also allows me to figure out any transitions. In some cases, I have paired this with nashville numbers to get a part right. All of this helps me to memorize a song so that I can play my part from memory without the other instruments present just by counting. I find, personally, that after playing the full song with the group, musical memory takes over and instead of having to count my way through a verse to know when the transition happens, I'll be hearing the other parts in my head as I'm playing. This, for me, is what makes it possible to memorize a large number of songs.
One thing that I might suggest, in your case, is to ask the band if they have any recordings of them playing the songs in question. Blues isn't complex, but there can be different arrangements for the instruments and the song itself (for instance, an extra verse, an extended vamp, etc). It'll be good to learn the general song, but if your band can provide you with past recordings of their arrangements, you can work using those as a base and avoid some surprises.
This is what I do. It works for me; it may not work for you. However, it's an option. Good luck!